Integrate the odes to check property P

Imports

from mpmath import mp
from mpmath import mpf,mpc
import sage.libs.mpmath.all as a
mp.pretty = True
from timeit import default_timer as timer
# start=timer()
# end=timer()

Functions to analyze the odes over a range of parameters

Each ode is parametrized by two numbers, $\sigma$ and $\alpha$ which are functions of

• $j$ and $\epsilon$ for odes ${\bf 3}_{j}$, ${\bf 2}_{j}$, and ${\bf 1}_{j}$
• $\epsilon$ for ode ${\bf 2}$.

function to analyze ode 3$_{j}$, 2$_{j}$, or 1$_{j}$ at $j=\infty$ for list of $\sigma$ values

The limit $j\rightarrow \infty$ is taken with $\sigma$ fixed.

def process_sigma_list():
    global jval,epsilonb,sigmaval,alphaval
    global data_asymp
    set_precision()
    initialize_ode()
    for sigma_arg in sigma_list:
        epsilonb=0.0
        set_Jacobi_parameters()
        sigmaval=myR(sigma_arg)
        jval=Infinity
        alphaval = myR(limit(alphaexp,j=jval))
        #
        analyze_monodromy()
        if GOOD:
            data[ode_name]['asymp'][sigmaval]=data_dict

function to analyze ode 3$_{j}$, 2$_{j}$, or 1$_{j}$ for list of $j$ values given $\epsilon$

def process_j_list():
    global jval,sigmaval,alphaval
    global data_j
    set_precision()
    initialize_ode()
    set_Jacobi_parameters()
    eps_string = epsilonb.str(digits=4,skip_zeroes=True)
    for j_arg in j_list:
        jval = j_arg
        alphaval = myR(alphaexp.subs(j=jval))
        sigmaval = myR(sigmaexp.subs(j=jval,epsb=myR(epsilonb)))
        #
        analyze_monodromy()
        if GOOD:
            if eps_string not in data[ode_name]['finite']:
                data[ode_name]['finite'][eps_string]={}
            data[ode_name]['finite'][eps_string][jval]=data_dict

Function to analyze ode 2 for list of $\epsilon$ values

def process_ode2_epsilonb_list():
    global jval,epsilonb,sigmaval,alphaval,ode_decimal_precision
    global data_small_j
    set_precision()
    initialize_ode()
    jval=None
    for epsilonb_arg in epsilonb_list:
        epsilonb=epsilonb_arg
        eps_string = epsilonb.str(digits=4,skip_zeroes=True)
        set_Jacobi_parameters()
        alphaval = myR(alphaexp)
        sigmaval = myR(sigmaexp.subs(epsb=myR(epsilonb)))
        #
        analyze_monodromy()
        if GOOD:
            data[ode_name][eps_string]=data_dict

Function to analyze imaginary period monodromy matrix

def analyze_monodromy():
    global data_dict,GOOD
    start = timer()
    GOOD=True
    if gauged:
        construct_gauge()
    mp.dps=ode_decimal_precision
    create_ode()
    start_ode=timer()
    calculate_Mi()
    end_ode=timer();
    ode_time = end_ode-start_ode
    mp.dps=high_decimal_precision
    check_Mi()
    if gauged:
        check_gauge_invariance()
        construct_Vphys()
    construct_Mi_phys()
    check_property_P()
    if property_P:
        calculate_frequencies()
        end=timer()
        total_time = end-start
        eps_string = epsilonb.str(digits=4,skip_zeroes=True)
        pretty_print(LE('\\epsilon ='), eps_string,LE('\\quad j ='), jval,\
            LE('\\quad\\sigma ='), n(sigmaval,digits=3), LE('\\quad(\\omega/\\sigma)^{2}='), mpstr(fs_ratio),\
                        '   (', ode_decimal_precision,', ', decimal_tol, ')', \
                     '    ode time = ', n(ode_time,digits=5), '   other time = ',  n(total_time-ode_time,digits=5) )
    if GOOD:
        data_dict = dictify_vars('jval','sigmaval','epsilonb','fs_ratio','Mi','Mi_phys','H','Vmat',\
                                 'ode_decimal_precision','high_decimal_precision',\
                    'N','gauged','H_eigenvalues','min_H_eigenvalue','positive_frequencies','negative_frequencies',\
                    'MiR','Mi_eigenvalues')

subroutines

def calculate_Mi():
    global Mi4,Mi2,Mi
    Mi4 = cP(Jac_Kp)
    Mi2 = cR*Omega*Mi4.transpose()*Omega*cR*Mi4
    Mi = Omega*Mi2.transpose_conj()*Omega*Mi2

def check_Mi():
    global GOOD
    if not aeq(mp.det(Mi),1):
        GOOD=False
        print ('FAIL: Mi does not have det 1')
    if not aeq(Mi.transpose()*Omega*Mi,Omega):
        GOOD=False
        print ('FAIL: Mi is not symplectic')

def check_gauge_invariance():
    global GOOD
    if not aeq(Wgauge,Mi*Wgauge): 
        GOOD = False
        print('Mi FAILS gauge invariance check') 

def construct_Vphys():
    global Vmat,GOOD
    Proj_rem = Proj_phys
    Nphys = N-1
    vlist=[]
    vplist=[]
    for ii in range(Nphys):
        v = mp.matrix(int(Crank),int(1))
        v[ii]=1
        vphys = Proj_rem*v
        vphys = (1/mp.norm(vphys))*vphys
        vpphys = mp.j*Omega*vphys
        vlist.append(vphys)
        vplist.append(vpphys)
        Proj_rem = Proj_rem - vphys*vphys.transpose()- vpphys*vpphys.transpose()
    Vmat = mp.matrix(int(Crank),int(0))
    for ii in range(Nphys):
        Vmat=mp.extend(Vmat,vlist[ii])
    for ii in range(Nphys):
        Vmat=mp.extend(Vmat,vplist[ii])
    if not aeq(Vmat.transpose()*Vmat,mp.eye(int(2*Nphys))):
        GOOD=False
        print('FAIL Vmat^t Vmat not = 1')

def construct_Mi_phys():
    global Mi_phys, Omega_phys,GOOD
    if gauged:
        Mi_phys = Vmat.transpose()*Mi*Vmat
        Omega_phys = Vmat.transpose()*Omega*Vmat
        if not aeq(mp.det(Mi_phys),1):
            GOOD=False
            print('FAIL Mi_phys not det 1')
        if not aeq(Mi_phys.transpose()*Omega_phys*Mi_phys,Omega_phys):
            GOOD=False
            print('FAIL Mi_phys not symplectic')
    else:
        Mi_phys = Mi
        Omega_phys = Omega

def check_property_P():
    global H,H_eigenvalues,property_P,min_H_eigenvalue,HR,GOOD
    H=Omega_phys*Mi_phys-Omega_phys
    H_eigenvalues_complex,HR = mp.eig(H)
    property_P = False
    if not aeq(H,H.transpose_conj()):
        GOOD=False
        print('H FAILS to be hermitian')
        return
    H_eigenvalues=[]
    for x in H_eigenvalues_complex:
        if a0(mp.im(x)):
            H_eigenvalues.append(mp.re(x))
        else:
            GOOD=False
            print('H FAILS with non-real eigenvalue')
    H_eigenvalues.sort()
    min_H_eigenvalue = H_eigenvalues[0]
    if min_H_eigenvalue > 0:
        property_P = True
    else:
        GOOD=False
        print('FAILS Property P')
    return

def calculate_frequencies():
    global Mi_phys,Mi_eigenvalues,positive_frequencies,negative_frequencies,min_frequency,MiR,GOOD,fs_ratio
    if not aeq(Mi_phys.transpose_conj()*H, H*Mi_phys):
        GOOD=False
        print('FAIL Mi_phys is not H-hermitian')
    Mi_eigenvalues_complex,MiR = mp.eig(Mi_phys)
    Mi_eigenvalues=[]
    for x in Mi_eigenvalues_complex:
        if a0(mp.im(x)):
            Mi_eigenvalues.append(mp.re(x))
        else:
            GOOD=False
            pretty_print('Mi FAILS: non-real eigenvalue')
    positive_frequencies =[]
    negative_frequencies =[]
    for x in Mi_eigenvalues:
        if x < 0:
            GOOD=False
            print('FAIL: Mi_phys negative eigenvalue')
        else:
            freq = mp.ln(x)/(4*Jac_Kp)
            if freq == 0:
                GOOD=False
                print('FAIL zero frequency')
            if freq < 0:
                negative_frequencies.append(freq)
            else:
                positive_frequencies.append(freq)
    if(len(negative_frequencies) != len(positive_frequencies)):
        GOOD=False
        print('FAIL Mi_phys number of positive and negative frequencies differ')
    negative_frequencies.sort(reverse=True)
    positive_frequencies.sort()
    for neg,pos in zip(negative_frequencies,positive_frequencies):
        if not aeq(-neg,pos):
            GOOD=False
            print('FAIL frequencies not in +- pairs')
    min_frequency = positive_frequencies[0]
    fs_ratio= (min_frequency/sigmaval)^2
    if N == 3:
        if not aeq(positive_frequencies[0],positive_frequencies[1]):
            print('DEGENERACY BROKEN')

subroutines for the ode solver

• initial condition at $\tau=0$ is $y_0=1$ the identity matrix
• K2mat, K1mat, K0mat are numerical matrices provided by Sagemath
• y_deriv(tau,y) calculates $\partial_\tau y = -i A(\tau) y$
• for the ode solver, the complex matrix $y$ is converted to a flat list of real numbers
• the solution mp.odefun(y_deriv ,0,y_init) is function of $\tau$ that returns a flat list.
• The propagator $P(\tau)$ is extracted from the solution function.

def create_ode():
    global cPlist,K0,K1,K2
    K0matval = K0mat.subs(alpha=alphaval)
    K2 = mpify_matrix(K2mat)
    K1 = mpify_matrix(K1mat*sigmaval)
    K0 = mpify_matrix(K0matval*sigmaval^2)
#  create ode
    cPlist = mp.odefun(y_deriv ,0,y_init,tol=ode_tol,degree=ode_degree)  

def construct_gauge():
    global Wgauge, Wpgauge, Proj_gauge, Proj_phys
    w0matval = w0mat.subs(alpha=alphaval)
    w1 = w1mat.change_ring(myC)
    w0 = (w0matval * sigmaval).change_ring(myC)
    zeroNvector = matrix(N,1)
    W0 = mpify_matrix(block_matrix(2,1,[w0,zeroNvector]))
    W1 = mpify_matrix(block_matrix(2,1,[w1,zeroNvector]))
    W1p = mpify_matrix(block_matrix(2,1,[zeroNvector,w1]))
    Wgauge = W0 + W1*F_at_K+ W1p*Fp_at_K
    Wpgauge = mp.j*Omega*Wgauge
    Proj_gauge = mp.eye(int(Crank)) - Wpgauge * mp.inverse(Wpgauge.transpose()*Wpgauge) * Wpgauge.transpose()
    Proj_phys = Proj_gauge - Wgauge* mp.inverse(Wgauge.transpose()*Wgauge) * Wgauge.transpose()

def y_deriv(tau,y):
# y is a list of length 4N * Crank
#     representing wr, wr', wi, wi'
# 
    Fi = -Jac_k * Jac_kp * mp.ellipfun('sd',u=tau,m=Jac_mp)
    Kr = K2*Fi^2 - K0
    Ny_4= N*Crank
    Ny_2= 2* Ny_4
    y1 = y[:Ny_2]
    y2 = y[Ny_2:]
    wr = matrixify_list(y1[:Ny_4],N,Crank)
    wpr = matrixify_list(y1[Ny_4:],N,Crank)
    wi = matrixify_list(y2[:Ny_4],N,Crank)
    wpi = matrixify_list(y2[Ny_4:],N,Crank)
    dwr = -wpi
    dwi = wpr
    dwpr = -Kr*wi+ Fi*(K1*wr)
    dwpi = Kr*wr + Fi*(K1*wi)
    dylist = listify_matrix(dwr) + listify_matrix(dwpr) + listify_matrix(dwi) + listify_matrix(dwpi)
    return dylist

def cP(tau):
    cPl = cPlist(tau)
    cPreal = mp.matrix([cPl[nn:nn+Crank] for nn in range(0,Crank*Crank,Crank)])
    cPimag = mp.matrix([cPl[nn:nn+Crank] for nn in range(Crank*Crank,Crank*Rrank,Crank)])
    cP_mp = cPreal+mp.j*cPimag
    return (cP_mp)

Initialization functions

def set_ranks():
    global Crank, Rrank, IN, ZN, Z2N, Omega, cR, y_init
    Crank = 2*N
    Rrank = 2*Crank
    IN = identity_matrix(Reals,N)
    ZN = matrix(Reals,N,N)
    I2N = identity_matrix(Reals,2*N)
    Z2N = matrix(Reals,2*N,2*N)
    Omega_sage = I*block_matrix([[ZN,IN],[-IN,ZN]])
    cR_sage= block_matrix([[IN,ZN],[ZN,-IN]])
    Omega = mpify_matrix(Omega_sage)
    cR = mpify_matrix(cR_sage)
    y_init_matrix_sage = block_matrix([[I2N],[Z2N]])
    y_init_matrix = mpify_matrix(y_init_matrix_sage)
    y_init = listify_matrix(y_init_matrix)

def initialize_ode():
    global ode,K2mat,K1mat,K0mat,K1coeff,K0coeff,alphaexp,sigmaexp,gauged,w1mat,w0mat,w0coeff,N
    ode = odes[ode_name]
    K2mat = ode['K2mat']
    K1mat = ode['K1mat']
    K0mat = ode['K0mat']
    alphaexp = ode['alphaexp']
    sigmaexp = ode['sigmaexp']
    gauged = ode['gauged']
    if gauged:
        w1mat = ode['w1mat']
        w0mat = ode['w0mat']
    N=K2mat.nrows()
    set_ranks()

Set precision and tolerances

• mp.dps and decimal_tol are set in the main code
• mp.dps is the decimal precision for mpmath.
• 10^{-decimal_tol} is the tolerance for approximate numerical equality.

The precision of Sagemath real and complex numbers is set to match the mpmath precision

def set_precision():
    global binary_precision,Reals, RealNumber, myR, Complexes, ComplexNumber, myC, I, J
    global mptol, ode_tol,ode_degree
#    mp.dps = decimal_precision
#    binary_precision = mp.prec
#
    binary_precision=mp.prec
    Reals = RealField(binary_precision)
    RealNumber = Reals
    myR = Reals
    Complexes = ComplexField(binary_precision)
    ComplexNumber = Complexes
    myC = Complexes
    I = myC(I)
#    set_tolerances
    mptol = mpf(10^(-decimal_tol))
    ode_tol = None
    ode_degree = None
#    print ('mp decimal precision =',decimal_precision, '    mp binary precision = sage binary precision =', binary_precision )

Set Jacobi parameters

def set_Jacobi_parameters():
    global epsilonb,Jac_m,Jac_mp,Jac_k,Jac_kp,Jac_K,Jac_Kp,F_at_K,Fp_at_K
    Jac_m = mpf(1/2) + mpf(epsilonb)^2
    Jac_mp = mpf(1)-Jac_m
    Jac_k = mp.sqrt(Jac_m)
    Jac_kp = mp.sqrt(Jac_mp)
    Jac_K = mp.ellipk(Jac_m)
    Jac_Kp = mp.ellipk(Jac_mp)
    F_at_K = 0.0
    Fp_at_K = - Jac_k* Jac_kp

Utilities

%display latex
def LE(str):
    return(LatexExpr(r''+ str))

def flatten_list(L):
    return [val for sublist in L for val in sublist]

def mpify_matrix(matrix_sage):
    return mp.matrix(list(map(list,list(matrix_sage))))

def sagify(x):
    return a.mpmath_to_sage(x,binary_precision)

def sagify_matrix(matrix_mp):
    return matrix([list(map(sagify,sublist)) for sublist in matrix_mp.tolist()])

def matrixify_list(ylist,nrows,ncols):
    return mp.matrix([[ylist[ii+jj] for jj in range(ncols)] for ii in range(0,nrows*ncols,ncols)])

def listify_matrix(mp_matrix):
    return flatten_list(mp_matrix.tolist())

print_digits = 20
def mpstr(x):
    if type(x) is list:
        return list(map(mpstr,x))
    return mp.nstr(x,print_digits)

dictify_vars = lambda *args: {i:eval(i) for i in args}

multby = lambda n,x: [n*item for item in x]

Numerical closeness utilities

$\text{mptol} = $ approximate equality tolerance $z\approx 0 \equiv |z| < \text{mptol}$

$\mathbf{a0(x)}\quad\mathrm{ iff \quad x.norm()\;<\;tol}{}^{2}$

aeq(x,y) iff $\mathrm{(x-y).norm}()\;<\mathrm{tol}^2$

clean(x)

• if $x$ real, if a0(x) set x=0
• if $x$ complex, set real and imaginary parts to 0 if a0
• otherwise assume $x$ is a matrix and clean its entries

def almosteq(x,y):
    return(mp.almosteq(x,y,mptol))
#
def aeq(x,y):
    if isinstance(x,mp.mpf) or isinstance(x,mp.mpc):
        return(almosteq(x,y))
    if isinstance(x,mp.matrix):
        return(almosteq(mp.norm(x-y),0))
def a0(x):
    if isinstance(x,mp.mpf) or isinstance(x,mp.mpc):
        return(almosteq(x,0))
    if isinstance(x,mp.matrix):
        return(almosteq(mp.norm(x),0))
def clean(x):
    if isinstance(x,mpf):
        y=x
        if a0(y):
            y = mpf(0)
        if aeq(y,1.0):
            y = mpf(1.0)
        if aeq(y,-1.0):
            y = mpf(-1.0)
        return y        
    if isinstance(x,mpc):
        xRe=clean(mp.re(x))
        xIm=clean(mp.im(x))
        if xIm == mpf(0):
            return(xRe)
        elif xRe == mpf(0):
            return(mp.j*xIm)
        else:
            return (xRe + mp.j* xIm)
    if type(x) is list:
        return(list(map(clean,x)))
    if isinstance(x,mp.matrix):
        xrows=x.rows
        xcols=x.cols
        y = mp.matrix(xrows,xcols)
        for i in range(xrows):
            for j in range(xcols):
                y[i,j]= clean(x[i,j])
        return y

Global variables

#     global Crank, Rrank, IN, ZN, Z2N, Omega, cR, y_init
#     global ode,K2mat,K1mat,K0mat,K1coeff,K0coeff,alphaexp,sigmaexp,gauged,w1mat,w0mat,w0coeff,N
#     global alpha_asymp, K0mat_asymp, w0mat_asymp
#     global binary_precision,Reals, RealNumber, myR, Complexes, ComplexNumber, myC, I, J
#     global epsilonb,Jac_m,Jac_mp,Jac_k,Jac_kp,Jac_K,Jac_Kp,F_at_t,Fp_at_t
#     global mptol, ode_tol,ode_degree
#     global cPlist,K0,K1,K2, Wgauge, Wpgauge, Proj_gauge, Proj_phys
#     global Mi4,Mi2,Mi
#     global GOOD
#     global Vmat,GOOD
#     global Mi_phys, Omega_phys,GOOD
#     global H,H_eigenvalues,property_P,min_H_eigenvalue,HR,GOOD
#     global Mi_phys,Mi_eigenvalues,positive_frequencies,negative_frequencies,min_frequency,MiR,GOOD

Check gauge invariance of $P(\tau)$

This can be used after a run, i.e. after an ode solution is found.

def F(tau):
    return(Jac_k* mp.ellipfun('cn',u=Jac_K+mp.j*tau,m=Jac_m))
def Fp(tau):
    return(- Jac_k* mp.ellipfun('sn',u=Jac_K+mp.j*tau,m=Jac_m)* mp.ellipfun('dn',u=Jac_K+mp.j*tau,m=Jac_m))
def Fi(tau):
    return(-Jac_k * Jac_kp * mp.ellipfun('sd',u=tau,m=Jac_mp))
def Wg(tau):
    w0matval = w0mat.subs(alpha=alphaval)
    w1 = w1mat.change_ring(myC)
    w0 = (w0matval * w0coeff.subs(sigma=sigmaval)).change_ring(myC)
    zeroNvector = matrix(N,1)
    W0 = mpify_matrix(block_matrix(2,1,[w0,zeroNvector]))
    W1 = mpify_matrix(block_matrix(2,1,[w1,zeroNvector]))
    W1p = mpify_matrix(block_matrix(2,1,[zeroNvector,w1]))
    return(W0 + W1*F(tau)+ W1p*Fp(tau))

Run the analyses

Integrating the ode requires a certain decimal precision in the ode solver.
For each calculation below the decimal precision has been set by trial and error.
Various consistency checks were made on each calculation.
The decimal precision was raised until all the consistency checks were passed.

load the odes, set precision and tolerance

odes=load('odes')
high_decimal_precision=300
mp.dps= high_decimal_precision
decimal_tol = 20
data = {}

Ode 2

ode_name='2'
data[ode_name]={}

ode_decimal_precision = 30
epsilonb_list = [0.001,0.002,0.003,0.004,0.005,0.01,0.02,0.03,0.04,0.05,0.1,0.2,0.3,0.4,0.5,0.6,0.7071]
process_ode2_epsilonb_list()

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.001 \quad j = \mathrm{None} \quad\sigma = 0.000707 \quad(\omega/\sigma)^{2}= 2.0 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.6006 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.76988\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.002 \quad j = \mathrm{None} \quad\sigma = 0.00141 \quad(\omega/\sigma)^{2}= 2.0 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.4197 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.27256\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.003 \quad j = \mathrm{None} \quad\sigma = 0.00212 \quad(\omega/\sigma)^{2}= 2.0 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.3831 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.27702\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.004 \quad j = \mathrm{None} \quad\sigma = 0.00283 \quad(\omega/\sigma)^{2}= 2.0 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.3885 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.27746\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.005 \quad j = \mathrm{None} \quad\sigma = 0.00354 \quad(\omega/\sigma)^{2}= 2.0 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.3839 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.27714\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = \mathrm{None} \quad\sigma = 0.00707 \quad(\omega/\sigma)^{2}= 2.0 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.3782 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.27009\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.02 \quad j = \mathrm{None} \quad\sigma = 0.0141 \quad(\omega/\sigma)^{2}= 2.0 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.3872 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.26741\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.03 \quad j = \mathrm{None} \quad\sigma = 0.0212 \quad(\omega/\sigma)^{2}= 2.0 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.3786 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.26892\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.04 \quad j = \mathrm{None} \quad\sigma = 0.0283 \quad(\omega/\sigma)^{2}= 2.0 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.3819 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.26734\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.05 \quad j = \mathrm{None} \quad\sigma = 0.0354 \quad(\omega/\sigma)^{2}= 2.0 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.3894 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.26634\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.1 \quad j = \mathrm{None} \quad\sigma = 0.0707 \quad(\omega/\sigma)^{2}= 2.0 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.3721 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.26822\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.2 \quad j = \mathrm{None} \quad\sigma = 0.141 \quad(\omega/\sigma)^{2}= 2.0 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.9470 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.26340\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.3 \quad j = \mathrm{None} \quad\sigma = 0.212 \quad(\omega/\sigma)^{2}= 2.0 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.8780 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.26140\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.4 \quad j = \mathrm{None} \quad\sigma = 0.283 \quad(\omega/\sigma)^{2}= 2.0 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.8319 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.25747\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.5 \quad j = \mathrm{None} \quad\sigma = 0.354 \quad(\omega/\sigma)^{2}= 2.0 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.3270 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.25333\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.6 \quad j = \mathrm{None} \quad\sigma = 0.424 \quad(\omega/\sigma)^{2}= 2.0 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.8495 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.24138\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.7071 \quad j = \mathrm{None} \quad\sigma = 0.500 \quad(\omega/\sigma)^{2}= 2.0 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.0870 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.19430\]

Ode $\mathbf{3_j}$

ode_name='3j'
data[ode_name]={}
data[ode_name]['asymp']={}
data[ode_name]['finite']={}

$j$ finite

epsilonb = 0.7071

ode_decimal_precision = 30
j_list = [3/2]
process_j_list()

ode_decimal_precision = 40
j_list = [2,5/2]
process_j_list()

ode_decimal_precision = 50
j_list = [3,7/2,4]
process_j_list()

ode_decimal_precision = 60
j_list = [9/2,5]
process_j_list()

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.7071 \quad j = \frac{3}{2} \quad\sigma = 0.816 \quad(\omega/\sigma)^{2}= 4.4999834016824557022 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.3931 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.33249\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.7071 \quad j = 2 \quad\sigma = 1.12 \quad(\omega/\sigma)^{2}= 5.1999897704981085583 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.3838 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.63626\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.7071 \quad j = \frac{5}{2} \quad\sigma = 1.41 \quad(\omega/\sigma)^{2}= 5.4999932868850748519 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.3771 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.62858\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.7071 \quad j = 3 \quad\sigma = 1.71 \quad(\omega/\sigma)^{2}= 5.6571381458966265831 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 7.4176 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.3600\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.7071 \quad j = \frac{7}{2} \quad\sigma = 2.00 \quad(\omega/\sigma)^{2}= 5.749996519696881626 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 7.4287 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.3529\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.7071 \quad j = 4 \quad\sigma = 2.29 \quad(\omega/\sigma)^{2}= 5.8095211363058142583 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 7.4138 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.3535\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.7071 \quad j = \frac{9}{2} \quad\sigma = 2.58 \quad(\omega/\sigma)^{2}= 5.849997883443329219 \phantom{\verb!xxx!}\verb|(| 60 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 14.721 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.7578\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.7071 \quad j = 5 \quad\sigma = 2.87 \quad(\omega/\sigma)^{2}= 5.8787861619461414304 \phantom{\verb!xxx!}\verb|(| 60 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 14.702 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.7476\]

epsilonb = 0.55

ode_decimal_precision = 30
j_list = [3/2,2]
process_j_list()

ode_decimal_precision = 40
j_list = [5/2,3,7/2]
process_j_list()

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.55 \quad j = \frac{3}{2} \quad\sigma = 0.635 \quad(\omega/\sigma)^{2}= 4.0119140651818690656 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.5293 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.33699\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.55 \quad j = 2 \quad\sigma = 0.870 \quad(\omega/\sigma)^{2}= 4.8782922295280631875 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.5297 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.33234\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.55 \quad j = \frac{5}{2} \quad\sigma = 1.10 \quad(\omega/\sigma)^{2}= 5.284176568320185352 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 9.1205 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.81482\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.55 \quad j = 3 \quad\sigma = 1.33 \quad(\omega/\sigma)^{2}= 5.5041350396067247139 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 9.1109 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.82106\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.55 \quad j = \frac{7}{2} \quad\sigma = 1.56 \quad(\omega/\sigma)^{2}= 5.6363382785629739775 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 9.1179 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.81759\]

epsilonb = 0.4

ode_decimal_precision = 30
j_list = [3/2,2,5/2]
process_j_list()

ode_decimal_precision = 40
j_list = [3,7/2]
process_j_list()

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.4 \quad j = \frac{3}{2} \quad\sigma = 0.462 \quad(\omega/\sigma)^{2}= 3.3043168365709716246 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.5847 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.34620\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.4 \quad j = 2 \quad\sigma = 0.632 \quad(\omega/\sigma)^{2}= 4.2926629349963078598 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.5846 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.34422\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.4 \quad j = \frac{5}{2} \quad\sigma = 0.800 \quad(\omega/\sigma)^{2}= 4.8595368961520156711 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.5844 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.34418\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.4 \quad j = 3 \quad\sigma = 0.966 \quad(\omega/\sigma)^{2}= 5.192260255551817162 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 10.749 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.84987\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.4 \quad j = \frac{7}{2} \quad\sigma = 1.13 \quad(\omega/\sigma)^{2}= 5.4001919850822057902 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 10.755 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.85234\]

epsilonb = 0.3

ode_decimal_precision = 30
j_list = [3/2,2,5/2,3]
process_j_list()

ode_decimal_precision = 40
j_list = [7/2]
process_j_list()

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.3 \quad j = \frac{3}{2} \quad\sigma = 0.346 \quad(\omega/\sigma)^{2}= 2.7500646261884110787 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.6443 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35779\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.3 \quad j = 2 \quad\sigma = 0.474 \quad(\omega/\sigma)^{2}= 3.6234231561114523133 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.6397 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.34864\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.3 \quad j = \frac{5}{2} \quad\sigma = 0.600 \quad(\omega/\sigma)^{2}= 4.2932101870569633592 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.6422 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35106\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.3 \quad j = 3 \quad\sigma = 0.725 \quad(\omega/\sigma)^{2}= 4.7452772824302272425 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.6379 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.34849\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.3 \quad j = \frac{7}{2} \quad\sigma = 0.849 \quad(\omega/\sigma)^{2}= 5.0484406379829194185 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 12.154 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.86802\]

epsilonb = 0.01

ode_decimal_precision = 30
j_list = [3/2,2,5/2,3,7/2,5,10,15,20,30,40,50,60,70,80,90]
process_j_list()

ode_decimal_precision = 40
j_list = [100,120,140,160]
process_j_list()

ode_decimal_precision = 50
j_list = [180,200,230]
process_j_list()

ode_decimal_precision = 70
j_list = [260,290,320,350]
process_j_list()

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = \frac{3}{2} \quad\sigma = 0.0115 \quad(\omega/\sigma)^{2}= 2.0006096917743312498 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2208 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35648\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 2 \quad\sigma = 0.0158 \quad(\omega/\sigma)^{2}= 2.001464161862771762 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2136 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35290\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = \frac{5}{2} \quad\sigma = 0.0200 \quad(\omega/\sigma)^{2}= 2.0025643102199197761 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2110 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35296\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 3 \quad\sigma = 0.0242 \quad(\omega/\sigma)^{2}= 2.0039112915562727295 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2180 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35330\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = \frac{7}{2} \quad\sigma = 0.0283 \quad(\omega/\sigma)^{2}= 2.005506513527785117 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2142 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.36143\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 5 \quad\sigma = 0.0406 \quad(\omega/\sigma)^{2}= 2.0117994436223473306 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2140 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35364\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 10 \quad\sigma = 0.0816 \quad(\omega/\sigma)^{2}= 2.049915088956064386 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2188 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35336\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 15 \quad\sigma = 0.122 \quad(\omega/\sigma)^{2}= 2.1173874742746105617 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2201 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35387\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 20 \quad\sigma = 0.163 \quad(\omega/\sigma)^{2}= 2.2191564847884813161 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2160 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35376\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 30 \quad\sigma = 0.245 \quad(\omega/\sigma)^{2}= 2.5408502878760358795 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2151 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35481\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 40 \quad\sigma = 0.327 \quad(\omega/\sigma)^{2}= 3.0023297892601524571 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2159 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35407\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 50 \quad\sigma = 0.408 \quad(\omega/\sigma)^{2}= 3.5145618548029226805 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2127 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35355\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 60 \quad\sigma = 0.490 \quad(\omega/\sigma)^{2}= 3.9836417374816104468 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2192 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35278\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 70 \quad\sigma = 0.572 \quad(\omega/\sigma)^{2}= 4.3691189358253653385 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2169 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35342\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 80 \quad\sigma = 0.653 \quad(\omega/\sigma)^{2}= 4.6713878684154581544 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2242 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35439\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 90 \quad\sigma = 0.735 \quad(\omega/\sigma)^{2}= 4.905275304219552021 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2145 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35925\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 100 \quad\sigma = 0.816 \quad(\omega/\sigma)^{2}= 5.086722917739205928 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 12.503 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.88602\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 120 \quad\sigma = 0.980 \quad(\omega/\sigma)^{2}= 5.3414664711113240511 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 12.492 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.88771\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 140 \quad\sigma = 1.14 \quad(\omega/\sigma)^{2}= 5.5053424306333022703 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 13.734 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.88781\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 160 \quad\sigma = 1.31 \quad(\omega/\sigma)^{2}= 5.615886790683154156 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 13.737 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.88991\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 180 \quad\sigma = 1.47 \quad(\omega/\sigma)^{2}= 5.6935841252232656642 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 29.015 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.0776\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 200 \quad\sigma = 1.63 \quad(\omega/\sigma)^{2}= 5.7501133827543736513 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 28.999 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.0672\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 230 \quad\sigma = 1.88 \quad(\omega/\sigma)^{2}= 5.8097230117170114877 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 31.901 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.0864\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 260 \quad\sigma = 2.12 \quad(\omega/\sigma)^{2}= 5.8504005667678360068 \phantom{\verb!xxx!}\verb|(| 70 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 109.08 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 8.3613\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 290 \quad\sigma = 2.37 \quad(\omega/\sigma)^{2}= 5.8793547844855359812 \phantom{\verb!xxx!}\verb|(| 70 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 109.11 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 8.3742\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 320 \quad\sigma = 2.61 \quad(\omega/\sigma)^{2}= 5.9006771964032691039 \phantom{\verb!xxx!}\verb|(| 70 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 109.09 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 8.3869\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 350 \quad\sigma = 2.86 \quad(\omega/\sigma)^{2}= 5.9168241864476698865 \phantom{\verb!xxx!}\verb|(| 70 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 109.09 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 8.3868\]

$j= \infty$

ode_decimal_precision = 30
sigma_list = [0.02,0.04,.06,.08,.10,0.15,0.2,0.25,0.3,0.35,0.4,0.45,0.5,0.55,.6,0.65,0.7,0.75]
process_sigma_list()

ode_decimal_precision = 40
sigma_list = [0.8,0.85,0.9,0.95,1,1.2]
process_sigma_list()

ode_decimal_precision = 50
sigma_list = [1.4,1.6,1.8]
process_sigma_list()

ode_decimal_precision = 60
sigma_list = [2.0,2.2,2.4,2.6]
process_sigma_list()

ode_decimal_precision = 70
sigma_list = [2.8,3.0]
process_sigma_list()

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.0200 \quad(\omega/\sigma)^{2}= 2.0029306396589592143 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2236 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.39343\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.0400 \quad(\omega/\sigma)^{2}= 2.01179637058130815 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2160 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35387\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.0600 \quad(\omega/\sigma)^{2}= 2.026815783726084496 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2137 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35326\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.0800 \quad(\omega/\sigma)^{2}= 2.0483431301388165799 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2211 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35313\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.100 \quad(\omega/\sigma)^{2}= 2.0768512508711160382 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2145 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35558\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.150 \quad(\omega/\sigma)^{2}= 2.1824734440870033519 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2211 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35398\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.200 \quad(\omega/\sigma)^{2}= 2.3443219375813765233 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2149 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35687\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.250 \quad(\omega/\sigma)^{2}= 2.5665348899140685796 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2161 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35399\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.300 \quad(\omega/\sigma)^{2}= 2.8416258944066673457 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2152 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35379\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.350 \quad(\omega/\sigma)^{2}= 3.1493685608081772441 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2148 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35361\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.400 \quad(\omega/\sigma)^{2}= 3.4640786507796314538 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2180 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35349\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.450 \quad(\omega/\sigma)^{2}= 3.7641093098968960934 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2153 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35513\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.500 \quad(\omega/\sigma)^{2}= 4.0364870439793698411 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2225 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35237\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.550 \quad(\omega/\sigma)^{2}= 4.2762245862832203116 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2150 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35806\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.600 \quad(\omega/\sigma)^{2}= 4.4835179731049804094 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2173 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35414\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.650 \quad(\omega/\sigma)^{2}= 4.6611734256132470388 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2189 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35309\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.700 \quad(\omega/\sigma)^{2}= 4.8129437327005195409 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2228 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35358\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.750 \quad(\omega/\sigma)^{2}= 4.9426438397638722761 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 5.2159 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.35461\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.800 \quad(\omega/\sigma)^{2}= 5.0537551161551944156 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 12.484 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.88884\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.850 \quad(\omega/\sigma)^{2}= 5.1492920034437529415 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 12.465 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.88868\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.900 \quad(\omega/\sigma)^{2}= 5.2317943046322504194 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 12.489 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.88517\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.950 \quad(\omega/\sigma)^{2}= 5.3033719781498695124 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 12.477 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.89113\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 1.00 \quad(\omega/\sigma)^{2}= 5.3657663950167112016 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 13.708 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.88630\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 1.20 \quad(\omega/\sigma)^{2}= 5.5486814297827637433 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 13.738 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.88871\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 1.40 \quad(\omega/\sigma)^{2}= 5.6635970315408634098 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 29.013 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.0731\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 1.60 \quad(\omega/\sigma)^{2}= 5.7400474766166610879 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 29.027 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.0897\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 1.80 \quad(\omega/\sigma)^{2}= 5.793310029131846546 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 31.963 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.0759\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 2.00 \quad(\omega/\sigma)^{2}= 5.831831372906549483 \phantom{\verb!xxx!}\verb|(| 60 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 59.779 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.4333\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 2.20 \quad(\omega/\sigma)^{2}= 5.8605596139979826243 \phantom{\verb!xxx!}\verb|(| 60 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 59.829 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.4454\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 2.40 \quad(\omega/\sigma)^{2}= 5.8825388177663555004 \phantom{\verb!xxx!}\verb|(| 60 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 65.649 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.4665\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 2.60 \quad(\omega/\sigma)^{2}= 5.8997208734549874499 \phantom{\verb!xxx!}\verb|(| 60 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 65.545 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.4212\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 2.80 \quad(\omega/\sigma)^{2}= 5.9134022698348587444 \phantom{\verb!xxx!}\verb|(| 70 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 108.59 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 8.3165\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 3.00 \quad(\omega/\sigma)^{2}= 5.9244706315060858988 \phantom{\verb!xxx!}\verb|(| 70 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 108.56 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 8.3171\]

Ode $\mathbf{2_j}$

ode_name='2j'
data[ode_name]={}
data[ode_name]['asymp']={}
data[ode_name]['finite']={}

$j$ finite

epsilonb = 0.7071

ode_decimal_precision = 30
j_list = [3/2]
process_j_list()

ode_decimal_precision = 40
j_list = [2,5/2]
process_j_list()

ode_decimal_precision = 50
j_list = [3,7/2,4]
process_j_list()

ode_decimal_precision = 60
j_list = [9/2,5]
process_j_list()

ode_decimal_precision = 70
j_list = [11/2]
process_j_list()

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.7071 \quad j = \frac{3}{2} \quad\sigma = 1.06 \quad(\omega/\sigma)^{2}= 3.111105996386973974 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.1663 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28558\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.7071 \quad j = 2 \quad\sigma = 1.41 \quad(\omega/\sigma)^{2}= 3.4999961970124620313 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.0246 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.52854\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.7071 \quad j = \frac{5}{2} \quad\sigma = 1.77 \quad(\omega/\sigma)^{2}= 3.6799973229185163812 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.0261 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.52670\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.7071 \quad j = 3 \quad\sigma = 2.12 \quad(\omega/\sigma)^{2}= 3.7777758319480891792 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 6.9061 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.2123\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.7071 \quad j = \frac{7}{2} \quad\sigma = 2.47 \quad(\omega/\sigma)^{2}= 3.8367332270254066081 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 6.9043 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.2123\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.7071 \quad j = 4 \quad\sigma = 2.83 \quad(\omega/\sigma)^{2}= 3.8749988587702415691 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 6.9113 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.2110\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.7071 \quad j = \frac{9}{2} \quad\sigma = 3.18 \quad(\omega/\sigma)^{2}= 3.9012336564667336 \phantom{\verb!xxx!}\verb|(| 60 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 14.025 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.5632\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.7071 \quad j = 5 \quad\sigma = 3.54 \quad(\omega/\sigma)^{2}= 3.9199992561534346762 \phantom{\verb!xxx!}\verb|(| 60 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 14.014 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.5638\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.7071 \quad j = \frac{11}{2} \quad\sigma = 3.89 \quad(\omega/\sigma)^{2}= 3.9338836793777357288 \phantom{\verb!xxx!}\verb|(| 70 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 25.137 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.7541\]

epsilonb = 0.6

ode_decimal_precision = 30
j_list = [3/2]
process_j_list()

ode_decimal_precision = 40
j_list = [2,5/2,3]
process_j_list()

ode_decimal_precision = 50
j_list = [7/2]
process_j_list()

ode_decimal_precision = 50
j_list = [4]
process_j_list()

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.6 \quad j = \frac{3}{2} \quad\sigma = 0.900 \quad(\omega/\sigma)^{2}= 3.0219757580714899217 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.6289 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.26299\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.6 \quad j = 2 \quad\sigma = 1.20 \quad(\omega/\sigma)^{2}= 3.4289121173565929182 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 7.1452 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.70020\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.6 \quad j = \frac{5}{2} \quad\sigma = 1.50 \quad(\omega/\sigma)^{2}= 3.628680571286386956 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 7.1397 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.69880\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.6 \quad j = 3 \quad\sigma = 1.80 \quad(\omega/\sigma)^{2}= 3.7400155842771586464 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 7.1452 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.69994\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.6 \quad j = \frac{7}{2} \quad\sigma = 2.10 \quad(\omega/\sigma)^{2}= 3.8080686609021688663 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 16.961 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.6797\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.6 \quad j = 4 \quad\sigma = 2.40 \quad(\omega/\sigma)^{2}= 3.8525999135959532319 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 16.974 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.6814\]

epsilonb = 0.5

ode_decimal_precision = 30
j_list = [3/2,2]
process_j_list()

ode_decimal_precision = 40
j_list = [5/2,3,7/2]
process_j_list()

ode_decimal_precision = 50
j_list = [9/2]
process_j_list()

ode_decimal_precision = 60
j_list = [6]
process_j_list()

ode_decimal_precision = 70
j_list = [8]
process_j_list()

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.5 \quad j = \frac{3}{2} \quad\sigma = 0.750 \quad(\omega/\sigma)^{2}= 2.9252106564059074814 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.1117 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.26875\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.5 \quad j = 2 \quad\sigma = 1.00 \quad(\omega/\sigma)^{2}= 3.336509692956312181 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.1070 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.26379\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.5 \quad j = \frac{5}{2} \quad\sigma = 1.25 \quad(\omega/\sigma)^{2}= 3.5574871095526227098 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 8.5732 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.72274\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.5 \quad j = 3 \quad\sigma = 1.50 \quad(\omega/\sigma)^{2}= 3.6859375596307213748 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 8.5624 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.72395\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.5 \quad j = \frac{7}{2} \quad\sigma = 1.75 \quad(\omega/\sigma)^{2}= 3.766268941358884203 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 8.5670 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.72235\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.5 \quad j = \frac{9}{2} \quad\sigma = 2.25 \quad(\omega/\sigma)^{2}= 3.8566326173765722934 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 20.328 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.7738\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.5 \quad j = 6 \quad\sigma = 3.00 \quad(\omega/\sigma)^{2}= 3.9186142347692286407 \phantom{\verb!xxx!}\verb|(| 60 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 42.169 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.8635\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.5 \quad j = 8 \quad\sigma = 4.00 \quad(\omega/\sigma)^{2}= 3.953986459962055927 \phantom{\verb!xxx!}\verb|(| 70 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 77.547 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 7.3441\]

epsilonb = 0.4

ode_decimal_precision = 30
j_list = [3/2,2,5/2]
process_j_list()

ode_decimal_precision = 40
j_list = [3,7/2,9/2]
process_j_list()

ode_decimal_precision = 50
j_list = [11/2]
process_j_list()

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.4 \quad j = \frac{3}{2} \quad\sigma = 0.600 \quad(\omega/\sigma)^{2}= 2.8260860543995007448 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.6120 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28126\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.4 \quad j = 2 \quad\sigma = 0.800 \quad(\omega/\sigma)^{2}= 3.2095538197436441631 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.6121 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.27748\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.4 \quad j = \frac{5}{2} \quad\sigma = 1.00 \quad(\omega/\sigma)^{2}= 3.448496411453397044 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.6040 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.27111\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.4 \quad j = 3 \quad\sigma = 1.20 \quad(\omega/\sigma)^{2}= 3.5985568104045214463 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 9.9381 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.74665\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.4 \quad j = \frac{7}{2} \quad\sigma = 1.40 \quad(\omega/\sigma)^{2}= 3.6966042108267113169 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 9.9436 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.74873\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.4 \quad j = \frac{9}{2} \quad\sigma = 1.80 \quad(\omega/\sigma)^{2}= 3.8107346628228260028 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 9.9419 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.74667\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.4 \quad j = \frac{11}{2} \quad\sigma = 2.20 \quad(\omega/\sigma)^{2}= 3.8713285468249709094 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 23.727 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.8277\]

epsilonb = 0.325

ode_decimal_precision = 30
j_list = [3/2,2,5/2]
process_j_list()

ode_decimal_precision = 40
j_list = [3,7/2]
process_j_list()

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.325 \quad j = \frac{3}{2} \quad\sigma = 0.487 \quad(\omega/\sigma)^{2}= 2.7686172645363766041 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.6586 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.27964\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.325 \quad j = 2 \quad\sigma = 0.650 \quad(\omega/\sigma)^{2}= 3.0910012237972445732 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.6586 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.27272\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.325 \quad j = \frac{5}{2} \quad\sigma = 0.812 \quad(\omega/\sigma)^{2}= 3.3289994780980640365 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.6567 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.27338\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.325 \quad j = 3 \quad\sigma = 0.975 \quad(\omega/\sigma)^{2}= 3.4945400530890411016 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 10.106 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.75690\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.325 \quad j = \frac{7}{2} \quad\sigma = 1.14 \quad(\omega/\sigma)^{2}= 3.6096034560197983957 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 10.097 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.75594\]

epsilonb = 0.25

ode_decimal_precision = 30
j_list = [3/2,2,5/2,3,7/2]
process_j_list()

ode_decimal_precision = 40
j_list = [9/2,13/2]
process_j_list()

ode_decimal_precision = 50
j_list = [8,10]
process_j_list()

ode_decimal_precision = 70
j_list = [23/2,13,29/2]
process_j_list()

ode_decimal_precision = 70
j_list = [16]
process_j_list()

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.25 \quad j = \frac{3}{2} \quad\sigma = 0.375 \quad(\omega/\sigma)^{2}= 2.7548735658800374599 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.2099 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.37515\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.25 \quad j = 2 \quad\sigma = 0.500 \quad(\omega/\sigma)^{2}= 2.9695797955370071617 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1040 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.27659\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.25 \quad j = \frac{5}{2} \quad\sigma = 0.625 \quad(\omega/\sigma)^{2}= 3.1718000990866994758 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1106 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.27571\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.25 \quad j = 3 \quad\sigma = 0.750 \quad(\omega/\sigma)^{2}= 3.3384548001964306367 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1044 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.27612\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.25 \quad j = \frac{7}{2} \quad\sigma = 0.875 \quad(\omega/\sigma)^{2}= 3.4682100918407403594 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1099 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.27591\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.25 \quad j = \frac{9}{2} \quad\sigma = 1.12 \quad(\omega/\sigma)^{2}= 3.6434974763842416896 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 10.253 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.76829\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.25 \quad j = \frac{13}{2} \quad\sigma = 1.62 \quad(\omega/\sigma)^{2}= 3.814084315501186252 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 11.384 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.76677\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.25 \quad j = 8 \quad\sigma = 2.00 \quad(\omega/\sigma)^{2}= 3.8739450433276470651 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 24.435 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.8778\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.25 \quad j = 10 \quad\sigma = 2.50 \quad(\omega/\sigma)^{2}= 3.917799937383960746 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 27.144 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.8812\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.25 \quad j = \frac{23}{2} \quad\sigma = 2.88 \quad(\omega/\sigma)^{2}= 3.937337767536398566 \phantom{\verb!xxx!}\verb|(| 70 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 93.658 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 7.9255\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.25 \quad j = 13 \quad\sigma = 3.25 \quad(\omega/\sigma)^{2}= 3.9506944761876452264 \phantom{\verb!xxx!}\verb|(| 70 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 93.619 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 7.9345\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.25 \quad j = \frac{29}{2} \quad\sigma = 3.62 \quad(\omega/\sigma)^{2}= 3.96021425385399209 \phantom{\verb!xxx!}\verb|(| 70 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 104.05 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 7.9300\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.25 \quad j = 16 \quad\sigma = 4.00 \quad(\omega/\sigma)^{2}= 3.9672316450013068105 \phantom{\verb!xxx!}\verb|(| 70 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 104.01 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 7.9302\]

epsilonb = 0.2

ode_decimal_precision = 30
j_list = [3/2,2,5/2,3,7/2]
process_j_list()

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.2 \quad j = \frac{3}{2} \quad\sigma = 0.300 \quad(\omega/\sigma)^{2}= 2.7900223637150635907 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1372 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.29612\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.2 \quad j = 2 \quad\sigma = 0.400 \quad(\omega/\sigma)^{2}= 2.911889776281244591 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1461 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.27750\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.2 \quad j = \frac{5}{2} \quad\sigma = 0.500 \quad(\omega/\sigma)^{2}= 3.0584348766204516843 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1318 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.27782\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.2 \quad j = 3 \quad\sigma = 0.600 \quad(\omega/\sigma)^{2}= 3.2022972174391099247 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1413 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28057\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.2 \quad j = \frac{7}{2} \quad\sigma = 0.700 \quad(\omega/\sigma)^{2}= 3.3295574084657802527 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1351 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.27788\]

epsilonb = 0.15

ode_decimal_precision = 30
j_list = [3/2,2,5/2,3,7/2]
process_j_list()

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.15 \quad j = \frac{3}{2} \quad\sigma = 0.225 \quad(\omega/\sigma)^{2}= 2.8767082020777589107 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1428 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28276\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.15 \quad j = 2 \quad\sigma = 0.300 \quad(\omega/\sigma)^{2}= 2.9060034398802915749 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1402 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28029\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.15 \quad j = \frac{5}{2} \quad\sigma = 0.375 \quad(\omega/\sigma)^{2}= 2.9730112551102726957 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1373 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28015\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.15 \quad j = 3 \quad\sigma = 0.450 \quad(\omega/\sigma)^{2}= 3.0622401264067358881 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1603 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28084\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.15 \quad j = \frac{7}{2} \quad\sigma = 0.525 \quad(\omega/\sigma)^{2}= 3.1595334734424986648 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1475 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28065\]

epsilonb = 0.125

ode_decimal_precision = 30
j_list = [3/2,2,5/2,3,7/2]
process_j_list()

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.125 \quad j = \frac{3}{2} \quad\sigma = 0.188 \quad(\omega/\sigma)^{2}= 2.9419804646675092648 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1639 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28233\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.125 \quad j = 2 \quad\sigma = 0.250 \quad(\omega/\sigma)^{2}= 2.933976756786076842 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1609 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.27968\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.125 \quad j = \frac{5}{2} \quad\sigma = 0.312 \quad(\omega/\sigma)^{2}= 2.9591701577989723021 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1634 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28209\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.125 \quad j = 3 \quad\sigma = 0.375 \quad(\omega/\sigma)^{2}= 3.0103722063219255875 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1644 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28125\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.125 \quad j = \frac{7}{2} \quad\sigma = 0.438 \quad(\omega/\sigma)^{2}= 3.0780823512890894891 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1616 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28042\]

epsilonb = 0.1

ode_decimal_precision = 30
j_list = [3/2,2,5/2,3,7/2,4]
process_j_list()

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.1 \quad j = \frac{3}{2} \quad\sigma = 0.150 \quad(\omega/\sigma)^{2}= 3.0202372938491234705 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1767 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28347\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.1 \quad j = 2 \quad\sigma = 0.200 \quad(\omega/\sigma)^{2}= 2.9874597291297524231 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1706 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28223\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.1 \quad j = \frac{5}{2} \quad\sigma = 0.250 \quad(\omega/\sigma)^{2}= 2.9771606541773666915 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1698 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28159\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.1 \quad j = 3 \quad\sigma = 0.300 \quad(\omega/\sigma)^{2}= 2.9892814480213505723 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1694 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28071\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.1 \quad j = \frac{7}{2} \quad\sigma = 0.350 \quad(\omega/\sigma)^{2}= 3.0200780860474762668 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1726 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28083\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.1 \quad j = 4 \quad\sigma = 0.400 \quad(\omega/\sigma)^{2}= 3.0645544106568858882 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1752 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28119\]

epsilonb = 0.08

ode_decimal_precision = 30
j_list = [3/2,2,5/2,3,7/2,4]
process_j_list()

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.08 \quad j = \frac{3}{2} \quad\sigma = 0.120 \quad(\omega/\sigma)^{2}= 3.0885824003150966159 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1877 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28467\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.08 \quad j = 2 \quad\sigma = 0.160 \quad(\omega/\sigma)^{2}= 3.0482652232348396391 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1732 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28186\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.08 \quad j = \frac{5}{2} \quad\sigma = 0.200 \quad(\omega/\sigma)^{2}= 3.0195927817201708685 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1777 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28124\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.08 \quad j = 3 \quad\sigma = 0.240 \quad(\omega/\sigma)^{2}= 3.0059180001580421244 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1736 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28072\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.08 \quad j = \frac{7}{2} \quad\sigma = 0.280 \quad(\omega/\sigma)^{2}= 3.0073934028957272796 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1771 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28022\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.08 \quad j = 4 \quad\sigma = 0.320 \quad(\omega/\sigma)^{2}= 3.022377590477805206 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1732 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28164\]

epsilonb = 0.01

ode_decimal_precision = 30
j_list = [3/2,2,5/2,3,7/2]+[m/2 for m in range(12,32,5)]+[m/2 for m in range(32,122,10)]
process_j_list()

j_list = [m/2 for m in range(122,182,20)]
process_j_list()

ode_decimal_precision = 40
j_list = [m/2 for m in range(182,342,20)]
process_j_list()

ode_decimal_precision = 50
j_list = [m/2 for m in range(342,502,40)]
process_j_list()

ode_decimal_precision = 60
j_list = [m/2 for m in range(500,660,80)]
process_j_list()

ode_decimal_precision = 80
j_list = [m/2 for m in range(660,820,80)]
process_j_list()

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = \frac{3}{2} \quad\sigma = 0.0150 \quad(\omega/\sigma)^{2}= 3.2672043864311006399 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1767 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28252\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 2 \quad\sigma = 0.0200 \quad(\omega/\sigma)^{2}= 3.2654202519516805471 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1784 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28043\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = \frac{5}{2} \quad\sigma = 0.0250 \quad(\omega/\sigma)^{2}= 3.263148361992893767 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1756 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28070\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 3 \quad\sigma = 0.0300 \quad(\omega/\sigma)^{2}= 3.2604047479759726416 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1721 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28314\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = \frac{7}{2} \quad\sigma = 0.0350 \quad(\omega/\sigma)^{2}= 3.2572085030256455228 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1723 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28140\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 6 \quad\sigma = 0.0600 \quad(\omega/\sigma)^{2}= 3.2352832046354860316 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1708 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28054\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = \frac{17}{2} \quad\sigma = 0.0850 \quad(\omega/\sigma)^{2}= 3.206121949782006774 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1733 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28038\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 11 \quad\sigma = 0.110 \quad(\omega/\sigma)^{2}= 3.1736417351061720374 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1783 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28086\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = \frac{27}{2} \quad\sigma = 0.135 \quad(\omega/\sigma)^{2}= 3.1414236275692625523 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1814 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28095\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 16 \quad\sigma = 0.160 \quad(\omega/\sigma)^{2}= 3.1122487572175275056 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1751 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28109\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 21 \quad\sigma = 0.210 \quad(\omega/\sigma)^{2}= 3.0696875586792866563 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1778 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28278\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 26 \quad\sigma = 0.260 \quad(\omega/\sigma)^{2}= 3.0521943360032247625 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1736 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28082\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 31 \quad\sigma = 0.310 \quad(\omega/\sigma)^{2}= 3.0584974343907985133 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1752 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28421\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 36 \quad\sigma = 0.360 \quad(\omega/\sigma)^{2}= 3.084063345537294108 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1753 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28403\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 41 \quad\sigma = 0.410 \quad(\omega/\sigma)^{2}= 3.1235085639426417268 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1712 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28119\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 46 \quad\sigma = 0.460 \quad(\omega/\sigma)^{2}= 3.1717877497744656857 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1730 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28000\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 51 \quad\sigma = 0.510 \quad(\omega/\sigma)^{2}= 3.2247105184655556095 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1740 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28046\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 56 \quad\sigma = 0.560 \quad(\omega/\sigma)^{2}= 3.2790876124698933971 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1738 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28120\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 61 \quad\sigma = 0.610 \quad(\omega/\sigma)^{2}= 3.3326732310851522941 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1720 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28033\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 71 \quad\sigma = 0.710 \quad(\omega/\sigma)^{2}= 3.4322462691108951995 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1708 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28018\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 81 \quad\sigma = 0.810 \quad(\omega/\sigma)^{2}= 3.5180450475770610144 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1753 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28111\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 91 \quad\sigma = 0.910 \quad(\omega/\sigma)^{2}= 3.5896614240309616849 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 11.596 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.78412\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 101 \quad\sigma = 1.01 \quad(\omega/\sigma)^{2}= 3.6486648274752725845 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 11.601 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.78210\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 111 \quad\sigma = 1.11 \quad(\omega/\sigma)^{2}= 3.6971296260447738647 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 11.588 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.78417\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 121 \quad\sigma = 1.21 \quad(\omega/\sigma)^{2}= 3.7370310384526100958 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 11.585 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.78203\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 131 \quad\sigma = 1.31 \quad(\omega/\sigma)^{2}= 3.7700511363266407975 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 11.586 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.78250\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 141 \quad\sigma = 1.41 \quad(\omega/\sigma)^{2}= 3.7975548447115829331 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 11.595 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.78294\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 151 \quad\sigma = 1.51 \quad(\omega/\sigma)^{2}= 3.820626139695773359 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 11.589 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.78194\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 161 \quad\sigma = 1.61 \quad(\omega/\sigma)^{2}= 3.8401187078847102487 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 11.588 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.78310\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 171 \quad\sigma = 1.71 \quad(\omega/\sigma)^{2}= 3.8567039682251523216 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 27.688 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.9285\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 191 \quad\sigma = 1.91 \quad(\omega/\sigma)^{2}= 3.8831601923182423238 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 27.697 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.9290\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 211 \quad\sigma = 2.11 \quad(\omega/\sigma)^{2}= 3.9030531012025266795 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 27.699 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.9312\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 231 \quad\sigma = 2.31 \quad(\omega/\sigma)^{2}= 3.9183462616756493358 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 27.693 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.9289\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 250 \quad\sigma = 2.50 \quad(\omega/\sigma)^{2}= 3.929798561076493345 \phantom{\verb!xxx!}\verb|(| 60 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 57.953 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.2429\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 290 \quad\sigma = 2.90 \quad(\omega/\sigma)^{2}= 3.9472798590403610717 \phantom{\verb!xxx!}\verb|(| 60 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 57.990 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.2475\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 330 \quad\sigma = 3.30 \quad(\omega/\sigma)^{2}= 3.9590035760635018684 \phantom{\verb!xxx!}\verb|(| 80 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 190.91 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 14.570\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 370 \quad\sigma = 3.70 \quad(\omega/\sigma)^{2}= 3.9672316604535058991 \phantom{\verb!xxx!}\verb|(| 80 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 190.33 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 14.593\]

$j\rightarrow \infty$

ode_decimal_precision = 30
sigma_list = [0.02,0.04,.06,.08,.10,0.15,0.2,0.25,0.3,0.35,0.4,0.45,0.5,0.55,.6,0.65,0.7,0.75,0.8,0.85,0.9]
process_sigma_list()

ode_decimal_precision = 40
sigma_list = [0.95,1,1.15,1.3,1.45,1.6]
process_sigma_list()

ode_decimal_precision = 50
sigma_list = [1.85,2.1]
process_sigma_list()

ode_decimal_precision = 60
sigma_list = [2.35,2.65,3.0]
process_sigma_list()

ode_decimal_precision = 70
sigma_list = [3.35,3.7]
process_sigma_list()

ode_decimal_precision = 80
sigma_list = [4.0]
process_sigma_list()

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.0200 \quad(\omega/\sigma)^{2}= 3.2668296476156010782 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1716 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.30129\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.0400 \quad(\omega/\sigma)^{2}= 3.2549624194185527516 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1589 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28258\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.0600 \quad(\omega/\sigma)^{2}= 3.2366200733195958741 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1549 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28074\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.0800 \quad(\omega/\sigma)^{2}= 3.2136259717282729338 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1632 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28002\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.100 \quad(\omega/\sigma)^{2}= 3.1880069613841409911 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1581 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28055\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.150 \quad(\omega/\sigma)^{2}= 3.1244685863857124095 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1562 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28007\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.200 \quad(\omega/\sigma)^{2}= 3.0771560571495292472 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1551 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28362\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.250 \quad(\omega/\sigma)^{2}= 3.0544570413357088867 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1496 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28029\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.300 \quad(\omega/\sigma)^{2}= 3.0561983284001380199 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1643 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28124\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.350 \quad(\omega/\sigma)^{2}= 3.0782494102199292586 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1518 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28037\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.400 \quad(\omega/\sigma)^{2}= 3.115268631675504591 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1571 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28013\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.450 \quad(\omega/\sigma)^{2}= 3.1620831000589164024 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1531 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28049\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.500 \quad(\omega/\sigma)^{2}= 3.2143090210928620561 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1573 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.27943\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.550 \quad(\omega/\sigma)^{2}= 3.2685555230236133845 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1517 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28112\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.600 \quad(\omega/\sigma)^{2}= 3.322397590023560884 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1528 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28011\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.650 \quad(\omega/\sigma)^{2}= 3.3742337839221701029 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1558 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28356\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.700 \quad(\omega/\sigma)^{2}= 3.4231048786692838874 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1519 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28016\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.750 \quad(\omega/\sigma)^{2}= 3.4685188101731642574 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1529 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.27987\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.800 \quad(\omega/\sigma)^{2}= 3.5103035433550040559 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1540 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28063\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.850 \quad(\omega/\sigma)^{2}= 3.5484937597379243632 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1566 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28041\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.900 \quad(\omega/\sigma)^{2}= 3.5832488584938412583 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.1546 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.28122\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.950 \quad(\omega/\sigma)^{2}= 3.614796434544753421 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 11.568 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.78447\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 1.00 \quad(\omega/\sigma)^{2}= 3.6433949315513066236 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 11.550 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.77963\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 1.15 \quad(\omega/\sigma)^{2}= 3.7141090034650305173 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 11.561 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.77976\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 1.30 \quad(\omega/\sigma)^{2}= 3.7670902207074880688 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 11.556 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.78179\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 1.45 \quad(\omega/\sigma)^{2}= 3.8073212784009466351 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 11.559 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.77998\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 1.60 \quad(\omega/\sigma)^{2}= 3.8383580870573511167 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 11.565 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.78105\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 1.85 \quad(\omega/\sigma)^{2}= 3.8760613017121425409 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 27.643 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.9265\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 2.10 \quad(\omega/\sigma)^{2}= 3.9022085114187509892 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 27.642 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.9286\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 2.35 \quad(\omega/\sigma)^{2}= 3.920997493443859379 \phantom{\verb!xxx!}\verb|(| 60 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 57.962 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.2299\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 2.65 \quad(\omega/\sigma)^{2}= 3.9372577317750071412 \phantom{\verb!xxx!}\verb|(| 60 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 57.927 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.2420\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 3.00 \quad(\omega/\sigma)^{2}= 3.9506514252909105507 \phantom{\verb!xxx!}\verb|(| 60 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 57.910 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 4.2339\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 3.35 \quad(\omega/\sigma)^{2}= 3.9602013952518077806 \phantom{\verb!xxx!}\verb|(| 70 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 106.41 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 8.1799\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 3.70 \quad(\omega/\sigma)^{2}= 3.9672405307184597691 \phantom{\verb!xxx!}\verb|(| 70 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 106.42 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 8.0976\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 4.00 \quad(\omega/\sigma)^{2}= 3.9718944720574996035 \phantom{\verb!xxx!}\verb|(| 80 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 190.40 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 14.553\]

Ode $\mathbf{1_j}$

ode_name='1j'
data[ode_name]={}
data[ode_name]['asymp']={}
data[ode_name]['finite']={}

$j$ finite

data['1j']['finite']={}

epsilonb = 0.7071

ode_decimal_precision = 40
j_list = [3/2]
process_j_list()

ode_decimal_precision = 40
j_list = [2,5/2]
process_j_list()

ode_decimal_precision = 50
j_list = [3,7/2,4]
process_j_list()

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.7071 \quad j = \frac{3}{2} \quad\sigma = 1.06 \quad(\omega/\sigma)^{2}= 4.4444480978367223919 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.5378 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.49725\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.7071 \quad j = 2 \quad\sigma = 1.41 \quad(\omega/\sigma)^{2}= 4.2500021920368275213 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.5364 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.41552\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.7071 \quad j = \frac{5}{2} \quad\sigma = 1.77 \quad(\omega/\sigma)^{2}= 4.1600014475720327977 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.5336 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.41471\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.7071 \quad j = 3 \quad\sigma = 2.12 \quad(\omega/\sigma)^{2}= 4.1111121340622449649 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 6.0651 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.0267\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.7071 \quad j = \frac{7}{2} \quad\sigma = 2.47 \quad(\omega/\sigma)^{2}= 4.0816334126785126723 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 6.0601 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.0277\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.7071 \quad j = 4 \quad\sigma = 2.83 \quad(\omega/\sigma)^{2}= 4.0625005856591875058 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 6.0614 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.0273\]

epsilonb = 0.5

ode_decimal_precision = 30
j_list = [3/2]
process_j_list()

ode_decimal_precision = 40
j_list = [2,5/2,3,7/2]
process_j_list()

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.5 \quad j = \frac{3}{2} \quad\sigma = 0.750 \quad(\omega/\sigma)^{2}= 4.6133268717304487007 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.6206 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.20663\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.5 \quad j = 2 \quad\sigma = 1.00 \quad(\omega/\sigma)^{2}= 4.3542164025695802042 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 7.6163 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.61383\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.5 \quad j = \frac{5}{2} \quad\sigma = 1.25 \quad(\omega/\sigma)^{2}= 4.2299812085082966298 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 7.6104 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.61240\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.5 \quad j = 3 \quad\sigma = 1.50 \quad(\omega/\sigma)^{2}= 4.1610829215236342351 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 7.6011 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.61191\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.5 \quad j = \frac{7}{2} \quad\sigma = 1.75 \quad(\omega/\sigma)^{2}= 4.1189953689480166178 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 7.6033 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.61291\]

epsilonb = 0.35

ode_decimal_precision = 30
j_list = [3/2,2,5/2]
process_j_list()

ode_decimal_precision = 40
j_list = [3,7/2]
process_j_list()

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.35 \quad j = \frac{3}{2} \quad\sigma = 0.525 \quad(\omega/\sigma)^{2}= 4.8909684915731806421 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.7455 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.21557\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.35 \quad j = 2 \quad\sigma = 0.700 \quad(\omega/\sigma)^{2}= 4.5358855153537569218 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.0874 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.21489\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.35 \quad j = \frac{5}{2} \quad\sigma = 0.875 \quad(\omega/\sigma)^{2}= 4.356661399443839756 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.0889 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.21478\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.35 \quad j = 3 \quad\sigma = 1.05 \quad(\omega/\sigma)^{2}= 4.2538454972933859456 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 8.9799 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.64676\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.35 \quad j = \frac{7}{2} \quad\sigma = 1.22 \quad(\omega/\sigma)^{2}= 4.1895656797028750557 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 8.9825 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.64468\]

epsilonb = 0.25

ode_decimal_precision = 30
j_list = [3/2,2,5/2,3,7/2]
process_j_list()

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.25 \quad j = \frac{3}{2} \quad\sigma = 0.375 \quad(\omega/\sigma)^{2}= 5.2882102889748398006 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.1448 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.21864\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.25 \quad j = 2 \quad\sigma = 0.500 \quad(\omega/\sigma)^{2}= 4.8130021114066852972 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.1451 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.21899\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.25 \quad j = \frac{5}{2} \quad\sigma = 0.625 \quad(\omega/\sigma)^{2}= 4.5586215534658206707 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.1446 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.21924\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.25 \quad j = 3 \quad\sigma = 0.750 \quad(\omega/\sigma)^{2}= 4.406506287046373958 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.1437 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.21951\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.25 \quad j = \frac{7}{2} \quad\sigma = 0.875 \quad(\omega/\sigma)^{2}= 4.3084606397713180209 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.1405 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.21886\]

epsilonb = 0.15

ode_decimal_precision = 30
j_list = [3/2,2,5/2,3,7/2,4]
process_j_list()

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.15 \quad j = \frac{3}{2} \quad\sigma = 0.225 \quad(\omega/\sigma)^{2}= 6.1901268599450224521 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.1731 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22467\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.15 \quad j = 2 \quad\sigma = 0.300 \quad(\omega/\sigma)^{2}= 5.5073555959279322088 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.1742 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22317\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.15 \quad j = \frac{5}{2} \quad\sigma = 0.375 \quad(\omega/\sigma)^{2}= 5.0997668285842154872 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.1732 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22212\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.15 \quad j = 3 \quad\sigma = 0.450 \quad(\omega/\sigma)^{2}= 4.8363596374252251442 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.1694 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22256\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.15 \quad j = \frac{7}{2} \quad\sigma = 0.525 \quad(\omega/\sigma)^{2}= 4.6563688999228258291 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.1696 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22223\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.15 \quad j = 4 \quad\sigma = 0.600 \quad(\omega/\sigma)^{2}= 4.5280529556201950957 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.1775 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22295\]

epsilonb = 0.01

ode_decimal_precision = 30
j_list = [3/2,2,5/2,3,7/2]+[m/2 for m in range(12,32,5)]+[m/2 for m in range(32,122,10)]
process_j_list()

j_list = [m/2 for m in range(122,182,20)]
process_j_list()

ode_decimal_precision = 40
j_list = [m/2 for m in range(182,342,20)]
process_j_list()

ode_decimal_precision = 50
j_list = [m/2 for m in range(342,502,40)]
process_j_list()

ode_decimal_precision = 60
j_list = [m/2 for m in range(500,660,80)]
process_j_list()

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = \frac{3}{2} \quad\sigma = 0.0150 \quad(\omega/\sigma)^{2}= 9.4738902559517118445 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.8419 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22532\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 2 \quad\sigma = 0.0200 \quad(\omega/\sigma)^{2}= 9.2413505144763454294 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.8459 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22291\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = \frac{5}{2} \quad\sigma = 0.0250 \quad(\omega/\sigma)^{2}= 9.1034872690655768504 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.8428 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22288\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 3 \quad\sigma = 0.0300 \quad(\omega/\sigma)^{2}= 8.9977300265488129387 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.8428 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22299\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = \frac{7}{2} \quad\sigma = 0.0350 \quad(\omega/\sigma)^{2}= 8.9032365967418369482 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.8485 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22264\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 6 \quad\sigma = 0.0600 \quad(\omega/\sigma)^{2}= 8.4348049879824903449 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.1984 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22315\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = \frac{17}{2} \quad\sigma = 0.0850 \quad(\omega/\sigma)^{2}= 7.9360626017987203662 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.1994 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22336\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 11 \quad\sigma = 0.110 \quad(\omega/\sigma)^{2}= 7.4521637269881348836 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.2116 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22350\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = \frac{27}{2} \quad\sigma = 0.135 \quad(\omega/\sigma)^{2}= 7.0148984974119558545 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.2009 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22274\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 16 \quad\sigma = 0.160 \quad(\omega/\sigma)^{2}= 6.634432658562039674 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.2008 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22309\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 21 \quad\sigma = 0.210 \quad(\omega/\sigma)^{2}= 6.0335279543213022425 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.2021 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22298\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 26 \quad\sigma = 0.260 \quad(\omega/\sigma)^{2}= 5.60129266793853753 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.2002 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22318\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 31 \quad\sigma = 0.310 \quad(\omega/\sigma)^{2}= 5.2869324427377732675 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.1996 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22247\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 36 \quad\sigma = 0.360 \quad(\omega/\sigma)^{2}= 5.0536364818784515874 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.2048 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22247\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 41 \quad\sigma = 0.410 \quad(\omega/\sigma)^{2}= 4.8767263549800293421 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.2008 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22284\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 46 \quad\sigma = 0.460 \quad(\omega/\sigma)^{2}= 4.7398231694763618516 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.1984 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22310\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 51 \quad\sigma = 0.510 \quad(\omega/\sigma)^{2}= 4.6319265908440108073 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.2012 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22282\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 56 \quad\sigma = 0.560 \quad(\omega/\sigma)^{2}= 4.5455044361387006678 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.2017 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22339\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 61 \quad\sigma = 0.610 \quad(\omega/\sigma)^{2}= 4.4752885985560094966 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.2008 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22392\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 71 \quad\sigma = 0.710 \quad(\omega/\sigma)^{2}= 4.3694500400463820245 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.2033 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22326\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 81 \quad\sigma = 0.810 \quad(\omega/\sigma)^{2}= 4.2948079405334089646 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.5615 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22297\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 91 \quad\sigma = 0.910 \quad(\omega/\sigma)^{2}= 4.2403263394586815659 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 9.3336 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.66989\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 101 \quad\sigma = 1.01 \quad(\omega/\sigma)^{2}= 4.1994225594625710714 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 9.3294 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.66906\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 111 \quad\sigma = 1.11 \quad(\omega/\sigma)^{2}= 4.167979915550613492 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 9.3277 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.66997\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 121 \quad\sigma = 1.21 \quad(\omega/\sigma)^{2}= 4.143320917670859077 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 10.360 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.66787\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 131 \quad\sigma = 1.31 \quad(\omega/\sigma)^{2}= 4.1236456759962382082 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 10.353 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.66899\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 141 \quad\sigma = 1.41 \quad(\omega/\sigma)^{2}= 4.1077096497099901994 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 10.365 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.66873\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 151 \quad\sigma = 1.51 \quad(\omega/\sigma)^{2}= 4.0946310478733501747 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 10.357 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.66948\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 161 \quad\sigma = 1.61 \quad(\omega/\sigma)^{2}= 4.0837716248010645245 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 10.359 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.67105\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 171 \quad\sigma = 1.71 \quad(\omega/\sigma)^{2}= 4.0746606294551170864 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 25.544 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.7394\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 191 \quad\sigma = 1.91 \quad(\omega/\sigma)^{2}= 4.0603558372035246335 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 25.599 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.7408\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 211 \quad\sigma = 2.11 \quad(\omega/\sigma)^{2}= 4.0497725139562661289 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 25.563 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.7378\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 231 \quad\sigma = 2.31 \quad(\omega/\sigma)^{2}= 4.0417303300317987103 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 25.579 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.7365\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 250 \quad\sigma = 2.50 \quad(\omega/\sigma)^{2}= 4.0357586472445873108 \phantom{\verb!xxx!}\verb|(| 60 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 54.514 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.9265\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.01 \quad j = 290 \quad\sigma = 2.90 \quad(\omega/\sigma)^{2}= 4.026722737877334427 \phantom{\verb!xxx!}\verb|(| 60 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 54.536 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.9360\]

ode_name='1j'
epsilonb = 0.001

ode_decimal_precision = 30
j_list = [3/2,2,5/2,3,7/2]+[m/2 for m in range(12,32,5)]+[m/2 for m in range(32,122,10)]
process_j_list()

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.001 \quad j = \frac{3}{2} \quad\sigma = 0.00150 \quad(\omega/\sigma)^{2}= 9.5276530065894383767 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.8454 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22449\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.001 \quad j = 2 \quad\sigma = 0.00200 \quad(\omega/\sigma)^{2}= 9.3328127565368508927 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.8473 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22280\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.001 \quad j = \frac{5}{2} \quad\sigma = 0.00250 \quad(\omega/\sigma)^{2}= 9.2423039854204874794 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.8417 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22456\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.001 \quad j = 3 \quad\sigma = 0.00300 \quad(\omega/\sigma)^{2}= 9.1927934511757878821 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.8401 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22253\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.001 \quad j = \frac{7}{2} \quad\sigma = 0.00350 \quad(\omega/\sigma)^{2}= 9.1625805849842362445 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.8437 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22271\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.001 \quad j = 6 \quad\sigma = 0.00600 \quad(\omega/\sigma)^{2}= 9.1033675328226328681 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.8393 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22283\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.001 \quad j = \frac{17}{2} \quad\sigma = 0.00850 \quad(\omega/\sigma)^{2}= 9.0812815323141515761 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.8400 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22288\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.001 \quad j = 11 \quad\sigma = 0.0110 \quad(\omega/\sigma)^{2}= 9.0648013382893590722 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.8414 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22287\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.001 \quad j = \frac{27}{2} \quad\sigma = 0.0135 \quad(\omega/\sigma)^{2}= 9.0484130879589703181 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.8438 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22296\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.001 \quad j = 16 \quad\sigma = 0.0160 \quad(\omega/\sigma)^{2}= 9.0305736661629697545 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.8391 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22338\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.001 \quad j = 21 \quad\sigma = 0.0210 \quad(\omega/\sigma)^{2}= 8.9887608911766755323 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.8449 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22384\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.001 \quad j = 26 \quad\sigma = 0.0260 \quad(\omega/\sigma)^{2}= 8.9381583304940496491 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.8475 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22353\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.001 \quad j = 31 \quad\sigma = 0.0310 \quad(\omega/\sigma)^{2}= 8.8790518944857559469 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.8438 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22301\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.001 \quad j = 36 \quad\sigma = 0.0360 \quad(\omega/\sigma)^{2}= 8.8121375510479175313 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.8420 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22303\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.001 \quad j = 41 \quad\sigma = 0.0410 \quad(\omega/\sigma)^{2}= 8.7382692060644393831 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.8434 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22302\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.001 \quad j = 46 \quad\sigma = 0.0460 \quad(\omega/\sigma)^{2}= 8.6583643093475697946 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.8423 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22316\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.001 \quad j = 51 \quad\sigma = 0.0510 \quad(\omega/\sigma)^{2}= 8.5733536534673498756 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.8476 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22334\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0.001 \quad j = 56 \quad\sigma = 0.0560 \quad(\omega/\sigma)^{2}= 8.4841488567930561233 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.1985 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22302\]

$j\rightarrow \infty$

ode_decimal_precision = 30
sigma_list = [0.001,0.01,0.02,0.04,.06,.08,.10,0.15,0.2,0.25,0.3,0.35,0.4,0.45,0.5,.6,0.7,0.8]
process_sigma_list()

ode_decimal_precision = 40
sigma_list = [0.9,1.00,1.15,1.30,1.45,1.6]
process_sigma_list()

ode_decimal_precision = 50
sigma_list = [1.85,2.10,2.35]
process_sigma_list()

ode_decimal_precision = 60
sigma_list = [2.65,3.0]
process_sigma_list()

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.00100 \quad(\omega/\sigma)^{2}= 9.0835330455416439274 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.8446 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22494\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.0100 \quad(\omega/\sigma)^{2}= 9.0612674877696019662 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.8430 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22265\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.0200 \quad(\omega/\sigma)^{2}= 8.9953627256328547181 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.8435 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22352\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.0400 \quad(\omega/\sigma)^{2}= 8.7529666561250607944 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 2.8420 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22419\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.0600 \quad(\omega/\sigma)^{2}= 8.4101046362495019905 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.1951 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22265\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.0800 \quad(\omega/\sigma)^{2}= 8.023039607122564902 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.1970 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22442\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.100 \quad(\omega/\sigma)^{2}= 7.6333997588370682807 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.1977 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22297\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.150 \quad(\omega/\sigma)^{2}= 6.7765482096795657426 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.1939 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22268\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.200 \quad(\omega/\sigma)^{2}= 6.1366900976602906555 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.1971 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22241\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.250 \quad(\omega/\sigma)^{2}= 5.6758107936233344452 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.1979 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22218\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.300 \quad(\omega/\sigma)^{2}= 5.3416107840948261744 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.2002 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22281\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.350 \quad(\omega/\sigma)^{2}= 5.0946036902475815238 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.2015 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22235\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.400 \quad(\omega/\sigma)^{2}= 4.9080698932375542945 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.1974 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22307\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.450 \quad(\omega/\sigma)^{2}= 4.7642709897178321575 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.1954 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22223\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.500 \quad(\omega/\sigma)^{2}= 4.6513274620791411213 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.1941 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22222\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.600 \quad(\omega/\sigma)^{2}= 4.4880550633673341101 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.1988 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22273\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.700 \quad(\omega/\sigma)^{2}= 4.3782629343472021188 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.3357 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.31863\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.800 \quad(\omega/\sigma)^{2}= 4.3011278478124393871 \phantom{\verb!xxx!}\verb|(| 30 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.7687 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.22601\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 0.900 \quad(\omega/\sigma)^{2}= 4.2450010384491445038 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 9.3235 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.67236\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 1.00 \quad(\omega/\sigma)^{2}= 4.2029703221284508332 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 9.3210 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.67393\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 1.15 \quad(\omega/\sigma)^{2}= 4.1573713863219759644 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 10.358 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.67006\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 1.30 \quad(\omega/\sigma)^{2}= 4.12539105593133652 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 10.365 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.67249\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 1.45 \quad(\omega/\sigma)^{2}= 4.1021461737375348904 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 10.355 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.67016\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 1.60 \quad(\omega/\sigma)^{2}= 4.0847481269085376936 \phantom{\verb!xxx!}\verb|(| 40 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 10.358 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 0.67153\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 1.85 \quad(\omega/\sigma)^{2}= 4.0641690184776466222 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 25.565 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.7412\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 2.10 \quad(\omega/\sigma)^{2}= 4.0502197825876173506 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 25.575 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.7443\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 2.35 \quad(\omega/\sigma)^{2}= 4.0403445880660037356 \phantom{\verb!xxx!}\verb|(| 50 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 25.599 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 1.7394\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 2.65 \quad(\omega/\sigma)^{2}= 4.0318918859941641937 \phantom{\verb!xxx!}\verb|(| 60 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 54.594 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.9360\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\epsilon = 0. \quad j = +\infty \quad\sigma = 3.00 \quad(\omega/\sigma)^{2}= 4.0249908291237605178 \phantom{\verb!xxx!}\verb|(| 60 \verb|,| 20 \verb|)| \phantom{\verb!xxxx!}\verb|ode|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 54.584 \phantom{\verb!xxx!}\verb|other|\phantom{\verb!x!}\verb|time|\phantom{\verb!x!}\verb|=| 3.9402\]

Graphics

ode_name='3j'
plot_data_j ={}
for eps_string,eps_data in data[ode_name]['finite'].items():
    if eps_string not in plot_data_j:
        plot_data_j[eps_string]=[]
    for jval,item in eps_data.items():
        sigmaval = item['sigmaval']
        pval = sqrt(6)*sigmaval
        p = RDF(pval)
        fs_ratio = RDF(item['fs_ratio']*sigmaval^2/pval^2)
        plot_data_j[eps_string].append((p,fs_ratio))
epsvals= list(plot_data_j.keys())
epsvals.sort()
print(epsvals)

['0.01', '0.3', '0.4', '0.55', '0.7071']

p=\
list_plot(plot_data_j['0.01'],legend_label=r'$\epsilon=.01$',marker='.',markersize=2,plotjoined=True,linestyle=":",thickness=0.5)\
+list_plot(plot_data_j['0.3'],legend_label=r'$\epsilon=.3$',marker='.',markersize=2,plotjoined=True,linestyle=":",thickness=0.5)\
+list_plot(plot_data_j['0.4'],legend_label=r'$\epsilon=.4$',marker='.',markersize=2,plotjoined=True,linestyle=":",thickness=0.5)\
+list_plot(plot_data_j['0.55'],legend_label=r'$\epsilon=.55$',marker='.',markersize=2,plotjoined=True,linestyle=":",thickness=0.5)\
+list_plot(plot_data_j['0.7071'],legend_label=r'$\epsilon=.7071$',marker='.',markersize=2,plotjoined=True,linestyle=":",thickness=0.5)\
+text(r"${\bf ode~3_j}$",(3,.6),fontsize=18,color='black')\
+line([(0,1),(7.2,1)],linestyle=":")
p.set_legend_options(loc='center right',markerscale=1,numpoints=1,shadow=False)
p.axes_labels(['$p$','$\\omega_{\\min}^2/p^2$'])
p.axes_labels_size(1.5)
p.show(ticks_integer=True,ymin=0.3)
p.save('3j_finite.pdf',dpi=600,ticks_integer=True,ymin=0.3)

ode_name='3j'
plot_data_asymp =[]
for key,item in data[ode_name]['asymp'].items():
    sigmaval = item['sigmaval']
    pval = sqrt(6)*sigmaval
    p = RDF(pval)
    fs_ratio = RDF(item['fs_ratio']*sigmaval^2/pval^2)
    plot_data_asymp.append((p,fs_ratio))

p=\
list_plot(plot_data_asymp,legend_label=r'$\epsilon=0$',marker=r'$\circ$',size=10)\
+list_plot(plot_data_j['0.01'],legend_label=r'$\epsilon=.01$',marker='.',size=8)\
+text(r"${\bf ode~3_j}$",(3,.6),fontsize=18,color='black')\
+line([(0,1),(7.5,1)],linestyle=":")
p.set_legend_options(loc='center right',markerscale=1,numpoints=1,shadow=False)
p.axes_labels(['$p$','$\\omega_{\\min}^2/p^2$'])
p.axes_labels_size(1.5)
p.show(ticks_integer=True,ymin=0.3)
p.save('3j_asymp.pdf',dpi=600,ticks_integer=True,ymin=0.3)

ode_name='2j'
plot_data_j ={}
for eps_string,eps_data in data[ode_name]['finite'].items():
    if eps_string not in plot_data_j:
        plot_data_j[eps_string]=[]
    for jval,item in eps_data.items():
        sigmaval = item['sigmaval']
        pval = 2*sigmaval
        p = RDF(pval)
        fs_ratio = RDF(item['fs_ratio']*sigmaval^2/pval^2)
        plot_data_j[eps_string].append((p,fs_ratio))
epsvals= list(plot_data_j.keys())
epsvals.sort()
print(epsvals)

['0.01', '0.08', '0.1', '0.125', '0.15', '0.2', '0.25', '0.325', '0.4', '0.5', '0.6', '0.7071']

p=\
list_plot(plot_data_j['0.01'],legend_label=r'$\epsilon=.01$',marker='.',markersize=2,plotjoined=True,linestyle=":",thickness=0.5)\
+list_plot(plot_data_j['0.08'],legend_label=r'$\epsilon=.08$',marker='.',markersize=2,plotjoined=True,linestyle=":",thickness=0.5)\
+list_plot(plot_data_j['0.1'],legend_label=r'$\epsilon=.1$',marker='.',markersize=2,plotjoined=True,linestyle=":",thickness=0.5)\
+list_plot(plot_data_j['0.125'],legend_label=r'$\epsilon=.125$',marker='.',markersize=2,plotjoined=True,linestyle=":",thickness=0.5)\
+list_plot(plot_data_j['0.15'],legend_label=r'$\epsilon=.15$',marker='.',markersize=2,plotjoined=True,linestyle=":",thickness=0.5)\
+list_plot(plot_data_j['0.2'],legend_label=r'$\epsilon=.2$',marker='.',markersize=2,plotjoined=True,linestyle=":",thickness=0.5)\
+list_plot(plot_data_j['0.25'],legend_label=r'$\epsilon=.25$',marker='.',markersize=2,plotjoined=True,linestyle=":",thickness=0.5)\
+list_plot(plot_data_j['0.325'],legend_label=r'$\epsilon=.325$',marker='.',markersize=2,plotjoined=True,linestyle=":",thickness=0.5)\
+list_plot(plot_data_j['0.4'],legend_label=r'$\epsilon=.4$',marker='.',markersize=2,plotjoined=True,linestyle=":",thickness=0.5)\
+list_plot(plot_data_j['0.5'],legend_label=r'$\epsilon=.5$',marker='.',markersize=2,plotjoined=True,linestyle=":",thickness=0.5)\
+list_plot(plot_data_j['0.6'],legend_label=r'$\epsilon=.6$',marker='.',markersize=2,plotjoined=True,linestyle=":",thickness=0.5)\
+list_plot(plot_data_j['0.7071'],legend_label=r'$\epsilon=.7071$',marker='.',markersize=2,plotjoined=True,linestyle=":",thickness=0.5)\
+text(r"${\bf ode~2_j}$",(4.5,.85),fontsize=18,color='black')\
+line([(0,1),(8,1)],linestyle=":")
p.set_legend_options(loc='center right',markerscale=1,numpoints=1,shadow=False)
p.axes_labels(['$p$','$\\omega_{\\min}^2/p^2$'])
p.axes_labels_size(1.5)
p.show(ticks_integer=True,xmin=0)
p.save('2j_finite.pdf',dpi=600,ticks_integer=True,xmin=0)

ode_name='2j'
plot_data_asymp =[]
for key,item in data[ode_name]['asymp'].items():
    sigmaval = item['sigmaval']
    pval = 2*sigmaval
    p = RDF(pval)
    fs_ratio = RDF(item['fs_ratio']*sigmaval^2/pval^2)
    plot_data_asymp.append((p,fs_ratio))

p=list_plot(plot_data_asymp,legend_label=r'$\epsilon=0$',marker=r'$\circ$',size=10)\
+list_plot(plot_data_j['0.01'],legend_label=r'$\epsilon=.01$',marker='.',size=8)\
+text(r"${\bf ode~2_j}$",(4.5,.85),fontsize=18,color='black')\
+line([(0,1),(8.0,1)],linestyle=":")
p.set_legend_options(loc='center right',markerscale=1,numpoints=1,shadow=False)
p.axes_labels(['$p$','$\\omega_{\\min}^2/p^2$'])
p.axes_labels_size(1.5)
p.show(ticks_integer=True)
p.save('2j_asymp.pdf',dpi=600,ticks_integer=True)

ode_name='1j'
plot_data_j ={}
for eps_string,eps_data in data[ode_name]['finite'].items():
    if eps_string not in plot_data_j:
        plot_data_j[eps_string]=[]
    for jval,item in eps_data.items():
        sigmaval = item['sigmaval']
        epsval = item['epsilonb']
        pval = sqrt(4*sigmaval^2+3*epsval^2)
        p = RDF(pval)
        fs_ratio = RDF(item['fs_ratio']*sigmaval^2/pval^2)
        plot_data_j[eps_string].append((p,fs_ratio))
epsvals= list(plot_data_j.keys())
epsvals.sort()
print(epsvals)

['0.001', '0.01', '0.15', '0.25', '0.35', '0.5', '0.7071']

p=\
list_plot(plot_data_j['0.01'],legend_label=r'$\epsilon=.01$',marker='.',markersize=2,plotjoined=True,linestyle=":",thickness=0.5)\
+list_plot(plot_data_j['0.15'],legend_label=r'$\epsilon=.15$',marker='.',markersize=2,plotjoined=True,linestyle=":",thickness=0.5)\
+list_plot(plot_data_j['0.25'],legend_label=r'$\epsilon=.25$',marker='.',markersize=2,plotjoined=True,linestyle=":",thickness=0.5)\
+list_plot(plot_data_j['0.35'],legend_label=r'$\epsilon=.35$',marker='.',markersize=2,plotjoined=True,linestyle=":",thickness=0.5)\
+list_plot(plot_data_j['0.5'],legend_label=r'$\epsilon=.5$',marker='.',markersize=2,plotjoined=True,linestyle=":",thickness=0.5)\
+list_plot(plot_data_j['0.7071'],legend_label=r'$\epsilon=.7071$',marker='.',markersize=2,plotjoined=True,linestyle=":",thickness=0.5)\
+text(r"${\bf ode~1_j}$",(2.5,1.5),fontsize=18,color='black')
p.set_legend_options(loc='center right',markerscale=1,numpoints=1,shadow=False)
p.axes_labels(['$p$','$\\omega_{\\min}^2/p^2$'])
p.axes_labels_size(1.5)
p.show(ticks_integer=True)
p.save('1j_finite.pdf',dpi=600,ticks_integer=True)

ode_name='1j'
plot_data_asymp =[]
for key,item in data[ode_name]['asymp'].items():
    sigmaval = item['sigmaval']
    pval = 2*sigmaval
#    pval = sqrt(4*sigmaval^2+3*epsval^2)
    p = RDF(pval)
    fs_ratio = RDF(item['fs_ratio']*sigmaval^2/pval^2)
    plot_data_asymp.append((p,fs_ratio))

p=list_plot(plot_data_asymp,legend_label=r'$\epsilon=0$',marker=r'$\circ$',size=10)\
+list_plot(plot_data_j['0.01'],legend_label=r'$\epsilon=.01$',marker='.',size=8)\
+text(r"${\bf ode~1_j}$",(2.5,1.5),fontsize=18,color='black')\
+line([(0,1),(6.0,1)],linestyle=":")
#p.set_legend_options(loc='center right',markerscale=1,numpoints=1,shadow=False)
p.axes_labels(['$p$','$\\omega_{\\min}^2/p^2$'])
p.set_legend_options(loc='center right',markerscale=1.5,shadow=False)
p.axes_labels_size(1.5)
p.show(ticks_integer=True,ymin=1)
p.save('1j_asymp.pdf',dpi=600,ticks_integer=True,ymin=1)

p=list_plot(plot_data_asymp,legend_label=r'$\epsilon=0$',marker=r'$\circ$',size=24)\
+list_plot(plot_data_j['0.001'],legend_label=r'$\epsilon=.001$',marker='.',markersize=3,plotjoined=True,thickness=0.5)\
+text(r"${\bf ode~1_j}$",(0.1,1.8),fontsize=18,color='black')\
+list_plot(plot_data_j['0.01'],legend_label=r'$\epsilon=.01$',marker='x',markersize=3,plotjoined=True,thickness=0.5)
p.axes_labels(['$p$','$\\omega_{\\min}^2/p^2$'])
p.set_legend_options(shadow=False)
#p.set_legend_options(loc='center right',markerscale=1.5,shadow=False)
p.axes_labels_size(1.5)
p.show(xmax=.2,ymin=1.7)
p.save('1j_smallp.pdf',dpi=600,xmax=.2,ymin=1.7)