Calculate $\mathbf{K}(z)$ matrices

$$ K(z) = K_2 b(z)^2 +K_1 b(z) + K_0 $$

Definitions and declarations

%display latex
maxima_calculus('algebraic: true;')
def LE(str):
    return(LatexExpr(r''+ str))

Racah W-coefficient

calculate W(j1,j2,J,j3,J12,J23,jtestmin='False'))

if j3 is a rational number

use the builtin racah() function

else

use the algebraic algorithm
return an algebraic expression
test the algebraic expression:

   if jtestmin is a rational number  
       substitute in the algebraic expression for W  
            jR = jtestmin, jtestmin+1/2, ... 11/2  
        and compare to racah()  

def W(j1,j2,J,j3,J12,J23,jtestmin='False'):
    if j3 in QQ:
        return(Wnum(j1,j2,J,j3,J12,J23))
    Walgval=Walg(j1,j2,J,j3,J12,J23)
    if jtestmin not in QQ:
        return(Walgval)
    for nn in range(2*jtestmin,12):
        jRval = nn/2
        Wnumval = Wnum(j1,j2,J.subs(jR=jRval),j3.subs(jR=jRval),J12,J23.subs(jR=jRval))
        Walgnumval=Walgval.subs(jR=jRval)
        if  Walgnumval not in QQ:
            Walgnumval=Walgnumval.canonicalize_radical()
        if Walgnumval != Wnumval:
            print('RACAH DISCREPANCY','jR=',jRval,'Wnum=',Wnumval,'Walgnum=',\
                  Walgnumval,'Walg=',Walgval)
        return(Walgval)

def Wnum(j1,j2,J,j3,J12,J23):
    Wnumval = racah(j1,j2,J,j3,J12,J23)
    if Wnumval not in QQ:
        Wnumval=Wnumval.canonicalize_radical()
    return(Wnumval)

def 𝚫sq(a,b,c):
    val = factorial(a+b-c)*factorial(a-b+c)*factorial(-a+b+c)/factorial(a+b+c+1)
    return val
def Walg(j1,j2,J,j3,J12,J23):
#    jR=var('jR',latex_name="j_{R}")
    assume(jR > 2)
    a=j1
    b=j2
    c=J
    d=j3
    e=J12
    f=J23
    𝞪1 = a+b+e
    𝞪2 = c+d+e
    𝞪3 = a+c+f
    𝞪4 = b+d+f
    𝞫1 = a+b+c+d
    𝞫2 = a+d+e+f
    𝞫3 = b+c+e+f
    max𝞪 = max([𝞪1,𝞪2,𝞪3,𝞪4])
    min𝞫 = min([𝞫1,𝞫2,𝞫3])
    w=0
    zrange=min𝞫 - max𝞪 + 1
    for zm in range(0,zrange):
        z=zm+max𝞪
        s = (-1)^(zm)*factorial(z+1)
        s = s/(factorial(z-𝞪1)*factorial(z-𝞪2)*factorial(z-𝞪3)*factorial(z-𝞪4))
        s= s/(factorial(𝞫1-z)*factorial(𝞫2-z)*factorial(𝞫3-z))
        w+=s
    d = 𝚫sq(a,b,e)*𝚫sq(c,d,e)*𝚫sq(a,c,f)*𝚫sq(b,d,f)
    d = d.full_simplify()
    Wval = w* sqrt(d)
    Wval = (((-1)^(𝞫1+max𝞪)*Wval).full_simplify()).subs((-1)^(4*jR)==1)
    Wval = Wval.canonicalize_radical()
    return(Wval)

def recoupling(j1,j2,J,j3,J12,J23,jtestmin='FALSE'):
    Wval= W(j1,j2,J,j3,J12,J23,jtestmin)
    recouplingval= sqrt((2*J12+1)*(2*J23+1))*Wval
#    recouplingval= recouplingval.canonicalize_radical()
    return(recouplingval)

Construct the matrices K_2, K_1, K_0

N, J12list, J23list, jL, jtestmin global variables

def constr_K_matrices():
    U = Matrix(SR,N,N)
    for ii in range(N):
        J12 = J12list[ii]
        for jj in range(N):
            J23 = J23list[jj]
            U[ii,jj] = recoupling(1,1,jR,jL,J12,J23,jtestmin)
    C12 = diagonal_matrix(SR, N, map(lambda x: x*(x+1)/2,J12list))
    C23diag = diagonal_matrix(SR, N, map(lambda x: x*(x+1)/2,J23list))
    C23 = U * C23diag * U.transpose()
    Gam = C12-2
    stard = 2*(C23- jR*(jR+1)/2)
    stardpG = stard+Gam
    K_2 = Gam^2-Gam
    K_1 = (stardpG*Gam+Gam*stardpG).canonicalize_radical()
    K_0 = (stardpG^2+Gam).canonicalize_radical()
    K_1 = K_1.subs(jR=jRsub).canonicalize_radical()
    K_0 = K_0.subs(jR=jRsub).canonicalize_radical()
    return(K_2,K_1,K_0)

epsb=var('epsb',latex_name="\\epsilon")
alpha=var('alpha',latex_name="\\alpha")
sigma=var('sigma',latex_name="\\sigma")
j = var('j')
odes={}

$\mathbf{ode}$ $\mathbf{2}$

N=2
J12list = [0,1]
J23list = [1/2,3/2]
jR=1/2
jL = 1/2
jtestmin=1/2
jRsub=j
(K_2,K_1,K_0) = constr_K_matrices()
pretty_print(LE('K_2 ='),K_2,LE('\\qquad K_1 ='),K_1,LE('\\epsilon_{b}'),\
             LE('\\qquad K_0 ='),K_0,LE('\\epsilon_{b}^{2}'))
#
print("\n")
alphaexp = 0
sigmaexp = epsb/sqrt(2)
K2mat = K_2
K1mat = K_1*sqrt(2)
K0mat = K_0*2
w1mat = Matrix([[0],[1]])
w0mat = Matrix([[1],[0]])
w0coeff = sigma/sqrt(2)
F= var('F')
Kmat= (K2mat*F^2+K1mat*sigma*F+K0mat*sigma^2).subs(alpha=alphaexp,sigma=sigmaexp)
wvec = (w1mat*F + w0mat*sigma).subs(alpha=alphaexp,sigma=sigmaexp)
#wvec = w1mat*F + w0mat*w0coeff
diff = w1mat *(2*epsb^2*F-2*F^3)+ Kmat*wvec
#diff = diff.canonicalize_radical()
#diff = diff.simplify_full()
#pretty_print(diff)
wconf = (diff == 0)
pretty_print(alpha,LE(' = '),alphaexp,LE('\\qquad'),sigma,LE(' = '),sigmaexp)
print("\n")
pretty_print(LE('K_2 ='),K2mat,LE('\\qquad K_1 = '),K1mat,\
             sigma,LE('\\qquad K_0 = '),K0mat,sigma^2)
print("\n")
#
#
print("\n")
pretty_print(LE('w_1 = '),w1mat,LE('\\qquad w_0 = '),w0mat,sigma,LE('\\qquad'),wconf)
#
ode={}
ode['K2mat']=K2mat
ode['K1mat']=K1mat
ode['K0mat']=K0mat
ode['alphaexp']=alphaexp
ode['sigmaexp']=sigmaexp
ode['w1mat']=w1mat
ode['w0mat']=w0mat
ode['gauged']=True
odes['2'] = ode

\[\newcommand{\Bold}[1]{\mathbf{#1}}K_2 = \left(\begin{array}{rr} 6 & 0 \\ 0 & 2 \end{array}\right) \qquad K_1 = \left(\begin{array}{rr} 0 & -3 \, \sqrt{2} \\ -3 \, \sqrt{2} & 0 \end{array}\right) \epsilon_{b} \qquad K_0 = \left(\begin{array}{rr} 0 & 0 \\ 0 & 1 \end{array}\right) \epsilon_{b}^{2}\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}{\alpha} = 0 \qquad {\sigma} = \frac{1}{2} \, \sqrt{2} {\epsilon}\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}K_2 = \left(\begin{array}{rr} 6 & 0 \\ 0 & 2 \end{array}\right) \qquad K_1 = \left(\begin{array}{rr} 0 & -6 \\ -6 & 0 \end{array}\right) {\sigma} \qquad K_0 = \left(\begin{array}{rr} 0 & 0 \\ 0 & 2 \end{array}\right) {\sigma}^{2}\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}w_1 = \left(\begin{array}{r} 0 \\ 1 \end{array}\right) \qquad w_0 = \left(\begin{array}{r} 1 \\ 0 \end{array}\right) {\sigma} \qquad \mathrm{True}\]

$\mathbf{ode}$ $\mathbf{3}_j$

jR=var('jR',latex_name="j_{R}")
N=3
J12list = [0,1,2]
J23list = [jR-1,jR,jR+1]
jL = jR
jtestmin=1
jRsub = j-1/2
(K_2,K_1,K_0) = constr_K_matrices()
pretty_print(LE('K_2 ='),K_2)
print("\n")
pretty_print(LE('K_1 ='),K_1,LE('\\epsilon_{b}'))
print("\n")
pretty_print(LE('K_0 ='),K_0,LE('\\epsilon_{b}^{2}'))
print("\n\n")
#
alphaexp = sqrt(2)* sqrt((j^2-1)/(j^2-1/4))
sigmaexp = sqrt(2/3)*sqrt(j^2-1/4)*epsb
#
K2mat = K_2
K1mat = -6*Matrix([[0,1,0],[1,0,0],[0,0,0]])
K0mat = 2*Matrix([[alpha^2,0,alpha],[0,alpha^2+1,0],[alpha,0,1]])
#
K1test = (sigma*K1mat).subs(alpha=alphaexp,sigma=sigmaexp)
K1test = K1test.canonicalize_radical()
K1eps = (K_1*epsb).canonicalize_radical()
conf1 = (K_1*epsb == K1test)
K0test = (sigma^2*K0mat).subs(alpha=alphaexp,sigma=sigmaexp)
K0test = K0test.canonicalize_radical()
K0eps = (K_0*epsb^2).canonicalize_radical()
conf0 = (K0eps == K0test)
F= var('F')
K = K_2 *F^2 + K_1*epsb*F +K_0*epsb^2
Kmat = (K2mat*F^2+K1mat*sigma*F+K0mat*sigma^2).subs(alpha=alphaexp,sigma=sigmaexp)
diff = (K-Kmat).canonicalize_radical()
conf = (diff == 0)
#
pretty_print(alpha,LE(' = '),alphaexp,LE('\\qquad'),sigma,LE(' = '),sigmaexp)
print("\n")
pretty_print(LE('K_2 ='),K2mat,LE('\\qquad K_1 = '),K1mat,sigma,LE('\\qquad K_0 = '),\
             K0mat,sigma^2,LE('\\qquad'),conf0 and conf1 and conf)
print("\n")
#
w1mat = Matrix([[0],[1],[0]])
w0mat = Matrix([[1],[0],[-alpha]])
wvec = (w1mat*F + w0mat*sigma).subs(alpha=alphaexp,sigma=sigmaexp)
diff = w1mat *(2*epsb^2*F-2*F^3)+ K*wvec
diff = diff.canonicalize_radical()
#diff = diff.simplify_full()
#pretty_print(diff)
wconf = (diff == 0)
pretty_print(LE('w_1 = '),w1mat,LE('\\qquad w_0 = '),w0mat,sigma,LE('\\qquad'),wconf)
#
#
ode={}
ode['K2mat']=K2mat
ode['K1mat']=K1mat
ode['K0mat']=K0mat
ode['alphaexp']=alphaexp
ode['sigmaexp']=sigmaexp
ode['w1mat']=w1mat
ode['w0mat']=w0mat
ode['gauged']=True
odes['3j'] = ode

\[\newcommand{\Bold}[1]{\mathbf{#1}}K_2 = \left(\begin{array}{rrr} 6 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 0 \end{array}\right)\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}K_1 = \left(\begin{array}{rrr} 0 & -\sqrt{3} \sqrt{2} \sqrt{2 \, j + 1} \sqrt{2 \, j - 1} & 0 \\ -\sqrt{3} \sqrt{2} \sqrt{2 \, j + 1} \sqrt{2 \, j - 1} & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) \epsilon_{b}\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}K_0 = \left(\begin{array}{rrr} \frac{8}{3} \, j^{2} - \frac{8}{3} & 0 & \frac{2}{3} \, \sqrt{2} \sqrt{2 \, j + 1} \sqrt{2 \, j - 1} \sqrt{j + 1} \sqrt{j - 1} \\ 0 & 4 \, j^{2} - 3 & 0 \\ \frac{2}{3} \, \sqrt{2} \sqrt{2 \, j + 1} \sqrt{2 \, j - 1} \sqrt{j + 1} \sqrt{j - 1} & 0 & \frac{4}{3} \, j^{2} - \frac{1}{3} \end{array}\right) \epsilon_{b}^{2}\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}{\alpha} = 2 \, \sqrt{2} \sqrt{\frac{j^{2} - 1}{4 \, j^{2} - 1}} \qquad {\sigma} = \frac{1}{2} \, \sqrt{\frac{2}{3}} \sqrt{4 \, j^{2} - 1} {\epsilon}\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}K_2 = \left(\begin{array}{rrr} 6 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 0 \end{array}\right) \qquad K_1 = \left(\begin{array}{rrr} 0 & -6 & 0 \\ -6 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) {\sigma} \qquad K_0 = \left(\begin{array}{rrr} 2 \, {\alpha}^{2} & 0 & 2 \, {\alpha} \\ 0 & 2 \, {\alpha}^{2} + 2 & 0 \\ 2 \, {\alpha} & 0 & 2 \end{array}\right) {\sigma}^{2} \qquad \mathrm{True}\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}w_1 = \left(\begin{array}{r} 0 \\ 1 \\ 0 \end{array}\right) \qquad w_0 = \left(\begin{array}{r} 1 \\ 0 \\ -{\alpha} \end{array}\right) {\sigma} \qquad \mathrm{True}\]

$\mathbf{ode}$ $\mathbf{2}_j$

jR=var('jR',latex_name="j_{R}")
j = var('j')
N=2
J12list = [1,2]
J23list = [jR,jR+1]
jL = jR+1
jtestmin=1/2
jRsub=j-1
(K_2,K_1,K_0) = constr_K_matrices()
pretty_print(LE('K_2 ='),K_2,LE('\\qquad K_1 ='),K_1,LE('\\epsilon_{b}'),\
             LE('\\qquad K_0 ='),K_0,LE('\\epsilon_{b}^{2}'))
print("\n\n")
#
alphaexp =  sqrt(1-1/j^2)
sigmaexp = j*epsb
K2mat=K_2
K1mat = 2*Matrix([[-1,0],[0,1]])
K0mat = 2*Matrix([[alpha^2,alpha],[alpha,1]])
#
K1test = (sigma*K1mat).subs(alpha=alphaexp,sigma=sigmaexp)
K1test = K1test.canonicalize_radical()
K1eps = (K_1*epsb).canonicalize_radical()
conf1 = (K_1*epsb == K1test)
K0test = (sigma^2*K0mat).subs(alpha=alphaexp,sigma=sigmaexp)
K0test = K0test.canonicalize_radical()
K0eps = (K_0*epsb^2).canonicalize_radical()
conf0 = (K0eps == K0test)
#
K = K_2 *F^2 + K_1*epsb*F +K_0*epsb^2
Kmat = (K2mat*F^2+K1mat*sigma*F+K0mat*sigma^2).subs(alpha=alphaexp,sigma=sigmaexp)
diff = (K-Kmat).canonicalize_radical()
conf = (diff == 0)
#
pretty_print(alpha,LE(' = '),alphaexp,LE('\\qquad'),sigma,LE(' = '),sigmaexp)
print("\n")
pretty_print(LE('K_2 ='),K2mat,LE('\\qquad K_1 = '),K1mat,sigma,LE('\\qquad K_0 = '),\
             K0mat,sigma^2,LE('\\qquad'),conf1 and conf0 and conf)
print("\n")
#
#
w1mat = Matrix([[1],[0]])
w0mat = Matrix([[1],[-alpha]])
Kmat= K_2*F^2+K_1*epsb*F+K_0*epsb^2
wvec = (w1mat*F + w0mat*sigma).subs(alpha=alphaexp,sigma=sigmaexp)
diff = w1mat *(2*epsb^2*F-2*F^3)+ Kmat*wvec
diff = diff.canonicalize_radical()
#diff = diff.simplify_full()
#pretty_print(diff)
wconf = (diff == 0)
pretty_print(LE('w_1 = '),w1mat,LE('\\qquad w_0 = '),w0mat,sigma,LE('\\qquad'),wconf)
#
ode={}
ode['K2mat']=K2mat
ode['K1mat']=K1mat
ode['K0mat']=K0mat
ode['alphaexp']=alphaexp
ode['sigmaexp']=sigmaexp
ode['w1mat']=w1mat
ode['w0mat']=w0mat
ode['gauged']=True
odes['2j'] = ode

\[\newcommand{\Bold}[1]{\mathbf{#1}}K_2 = \left(\begin{array}{rr} 2 & 0 \\ 0 & 0 \end{array}\right) \qquad K_1 = \left(\begin{array}{rr} -2 \, j & 0 \\ 0 & 2 \, j \end{array}\right) \epsilon_{b} \qquad K_0 = \left(\begin{array}{rr} 2 \, j^{2} - 2 & 2 \, \sqrt{j + 1} \sqrt{j - 1} j \\ 2 \, \sqrt{j + 1} \sqrt{j - 1} j & 2 \, j^{2} \end{array}\right) \epsilon_{b}^{2}\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}{\alpha} = \sqrt{-\frac{1}{j^{2}} + 1} \qquad {\sigma} = {\epsilon} j\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}K_2 = \left(\begin{array}{rr} 2 & 0 \\ 0 & 0 \end{array}\right) \qquad K_1 = \left(\begin{array}{rr} -2 & 0 \\ 0 & 2 \end{array}\right) {\sigma} \qquad K_0 = \left(\begin{array}{rr} 2 \, {\alpha}^{2} & 2 \, {\alpha} \\ 2 \, {\alpha} & 2 \end{array}\right) {\sigma}^{2} \qquad \mathrm{True}\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}w_1 = \left(\begin{array}{r} 1 \\ 0 \end{array}\right) \qquad w_0 = \left(\begin{array}{r} 1 \\ -{\alpha} \end{array}\right) {\sigma} \qquad \mathrm{True}\]

$\mathbf{ode}$ $\mathbf{1}_j$

jR=var('jR',latex_name="j_{R}")
j = var('j')
N=1
J12list = [2]
J23list = [jR+1]
jL=jR+2
jtestmin=1/2
jRsub=j-3/2
(K_2,K_1,K_0) = constr_K_matrices()
pretty_print(LE('K_2 ='),K_2,LE('\\qquad K_1 ='),K_1,LE('\\epsilon_{b}'),\
             LE('\\qquad K_0 ='),K_0,LE('\\epsilon_{b}^{2}'))
print("\n")
#
sigmaexp = j*epsb
alphaexp = sqrt(1+1/(4*j^2))
K2mat = K_2
K1mat = 4 *Matrix([1])
K0mat = 4*Matrix([alpha^2])
#
K1test = (sigma*K1mat).subs(sigma=sigmaexp,alpha=alphaexp)
K1test = K1test.canonicalize_radical()
K1eps = (K_1*epsb).canonicalize_radical()
conf1 = (K_1*epsb == K1test)
K0test = (sigma^2*K0mat).subs(sigma=sigmaexp,alpha=alphaexp)
K0test = K0test.canonicalize_radical()
K0eps = (K_0*epsb^2).canonicalize_radical()
conf0 = (K0eps == K0test)
#
K = K_2 *F^2 + K_1*epsb*F +K_0*epsb^2
Kmat = (K2mat*F^2+K1mat*sigma*F+K0mat*sigma^2).subs(alpha=alphaexp,sigma=sigmaexp)
diff = (K-Kmat).canonicalize_radical()
conf = (diff == 0)
#
pretty_print(alpha,LE(' = '),alphaexp,LE('\\qquad'),sigma,LE(' = '),sigmaexp)
print("\n")
pretty_print(LE('K_2 ='),K2mat,LE('\\quad K_1 = '),K1mat,sigma,LE('\\qquad K_0 = '),\
             K0mat,sigma^2,LE('\\qquad'),conf0 and conf1 and conf)
#
ode={}
ode['K2mat']=K2mat
ode['K1mat']=K1mat
ode['K0mat']=K0mat
ode['alphaexp']=alphaexp
ode['sigmaexp']=sigmaexp
ode['gauged']=False
odes['1j'] = ode

\[\newcommand{\Bold}[1]{\mathbf{#1}}K_2 = \left(\begin{array}{r} 0 \end{array}\right) \qquad K_1 = \left(\begin{array}{r} 4 \, j \end{array}\right) \epsilon_{b} \qquad K_0 = \left(\begin{array}{r} 4 \, j^{2} + 1 \end{array}\right) \epsilon_{b}^{2}\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}{\alpha} = \frac{1}{2} \, \sqrt{\frac{1}{j^{2}} + 4} \qquad {\sigma} = {\epsilon} j\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}K_2 = \left(\begin{array}{r} 0 \end{array}\right) \quad K_1 = \left(\begin{array}{r} 4 \end{array}\right) {\sigma} \qquad K_0 = \left(\begin{array}{r} 4 \, {\alpha}^{2} \end{array}\right) {\sigma}^{2} \qquad \mathrm{True}\]

save(odes,'odes')  # load with odes=load('odes')

odes =load('odes')

for ode_key in ['2','3j','2j','1j']:
    print('\n')
    pretty_print(LE(r"\mathbf{ODE}\;\; "),ode_key)
    for (key,value) in odes[ode_key].items():
        pretty_print(key,' = ',value)

\[\newcommand{\Bold}[1]{\mathbf{#1}}\mathbf{ODE}\;\; 2\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\verb|K2mat| \phantom{\verb!x!}\verb|=| \left(\begin{array}{rr} 6 & 0 \\ 0 & 2 \end{array}\right)\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\verb|K1mat| \phantom{\verb!x!}\verb|=| \left(\begin{array}{rr} 0 & -6 \\ -6 & 0 \end{array}\right)\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\verb|K0mat| \phantom{\verb!x!}\verb|=| \left(\begin{array}{rr} 0 & 0 \\ 0 & 2 \end{array}\right)\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\verb|alphaexp| \phantom{\verb!x!}\verb|=| 0\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\verb|sigmaexp| \phantom{\verb!x!}\verb|=| \frac{1}{2} \, \sqrt{2} {\epsilon}\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\verb|w1mat| \phantom{\verb!x!}\verb|=| \left(\begin{array}{r} 0 \\ 1 \end{array}\right)\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\verb|w0mat| \phantom{\verb!x!}\verb|=| \left(\begin{array}{r} 1 \\ 0 \end{array}\right)\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\verb|gauged| \phantom{\verb!x!}\verb|=| \mathrm{True}\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\mathbf{ODE}\;\; \verb|3j|\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\verb|K2mat| \phantom{\verb!x!}\verb|=| \left(\begin{array}{rrr} 6 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 0 \end{array}\right)\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\verb|K1mat| \phantom{\verb!x!}\verb|=| \left(\begin{array}{rrr} 0 & -6 & 0 \\ -6 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\verb|K0mat| \phantom{\verb!x!}\verb|=| \left(\begin{array}{rrr} 2 \, {\alpha}^{2} & 0 & 2 \, {\alpha} \\ 0 & 2 \, {\alpha}^{2} + 2 & 0 \\ 2 \, {\alpha} & 0 & 2 \end{array}\right)\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\verb|alphaexp| \phantom{\verb!x!}\verb|=| 2 \, \sqrt{2} \sqrt{\frac{j^{2} - 1}{4 \, j^{2} - 1}}\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\verb|sigmaexp| \phantom{\verb!x!}\verb|=| \frac{1}{2} \, \sqrt{\frac{2}{3}} \sqrt{4 \, j^{2} - 1} {\epsilon}\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\verb|w1mat| \phantom{\verb!x!}\verb|=| \left(\begin{array}{r} 0 \\ 1 \\ 0 \end{array}\right)\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\verb|w0mat| \phantom{\verb!x!}\verb|=| \left(\begin{array}{r} 1 \\ 0 \\ -{\alpha} \end{array}\right)\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\verb|gauged| \phantom{\verb!x!}\verb|=| \mathrm{True}\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\mathbf{ODE}\;\; \verb|2j|\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\verb|K2mat| \phantom{\verb!x!}\verb|=| \left(\begin{array}{rr} 2 & 0 \\ 0 & 0 \end{array}\right)\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\verb|K1mat| \phantom{\verb!x!}\verb|=| \left(\begin{array}{rr} -2 & 0 \\ 0 & 2 \end{array}\right)\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\verb|K0mat| \phantom{\verb!x!}\verb|=| \left(\begin{array}{rr} 2 \, {\alpha}^{2} & 2 \, {\alpha} \\ 2 \, {\alpha} & 2 \end{array}\right)\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\verb|alphaexp| \phantom{\verb!x!}\verb|=| \sqrt{-\frac{1}{j^{2}} + 1}\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\verb|sigmaexp| \phantom{\verb!x!}\verb|=| {\epsilon} j\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\verb|w1mat| \phantom{\verb!x!}\verb|=| \left(\begin{array}{r} 1 \\ 0 \end{array}\right)\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\verb|w0mat| \phantom{\verb!x!}\verb|=| \left(\begin{array}{r} 1 \\ -{\alpha} \end{array}\right)\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\verb|gauged| \phantom{\verb!x!}\verb|=| \mathrm{True}\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\mathbf{ODE}\;\; \verb|1j|\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\verb|K2mat| \phantom{\verb!x!}\verb|=| \left(\begin{array}{r} 0 \end{array}\right)\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\verb|K1mat| \phantom{\verb!x!}\verb|=| \left(\begin{array}{r} 4 \end{array}\right)\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\verb|K0mat| \phantom{\verb!x!}\verb|=| \left(\begin{array}{r} 4 \, {\alpha}^{2} \end{array}\right)\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\verb|alphaexp| \phantom{\verb!x!}\verb|=| \frac{1}{2} \, \sqrt{\frac{1}{j^{2}} + 4}\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\verb|sigmaexp| \phantom{\verb!x!}\verb|=| {\epsilon} j\]

\[\newcommand{\Bold}[1]{\mathbf{#1}}\verb|gauged| \phantom{\verb!x!}\verb|=| \mathrm{False}\]

#pretty_print(odes['2'])

#pretty_print(odes['3j'])

#pretty_print(odes['2j'])

#pretty_print(odes['1j'])