反射透射系数矩阵详细表达式

Author: Zhu Dengda
Email: zhudengda@mail.iggcas.ac.cn

[1]:

import numpy as np
np.set_printoptions(linewidth=200)

import sympy as sp
from sympy.printing.latex import latex
from IPython.display import display, Math

[2]:

# 定义基本变量
k, b1, a1, mu1, Omg1, kb1, ka1, w = sp.symbols(r'k b_1 a_1 mu_1 \Omega_1 k_{b1} k_{a1} \omega')
b2, a2, mu2, Omg2, kb2, ka2 = sp.symbols(r'b_2 a_2 mu_2 \Omega_2 k_{b2} k_{a2}')
lam1, lam2 = sp.symbols(r'\lambda_1 \lambda_2')
rho1, rho2 = sp.symbols(r'\rho_1 \rho_2')

原推导方法

[3]:

Ds1 = sp.Matrix([
    [k,            b1,             k,            -b1],
    [a1,            k,             -a1,           k],
    [2*mu1*Omg1,     2*k*mu1*b1,      2*mu1*Omg1,     -2*k*mu1*b1],
    [2*k*mu1*a1,     2*mu1*Omg1,      -2*k*mu1*a1,    2*mu1*Omg1]
])

Ds2 = sp.Matrix([
    [k,            b2,             k,            -b2],
    [a2,            k,             -a2,           k],
    [2*mu2*Omg2,     2*k*mu2*b2,      2*mu2*Omg2,     -2*k*mu2*b2],
    [2*k*mu2*a2,     2*mu2*Omg2,      -2*k*mu2*a2,    2*mu2*Omg2]
])

Q = sp.together(sp.expand(Ds1.inv() * Ds2))
Q

[3]:

$\displaystyle \left[\begin{matrix}\frac{- \Omega_{1} a_{2} \mu_{1} - \Omega_{2} a_{1} \mu_{2} + a_{1} k^{2} \mu_{1} + a_{2} k^{2} \mu_{2}}{2 a_{1} \mu_{1} \left(- \Omega_{1} + k^{2}\right)} & \frac{k \left(- \Omega_{1} \mu_{1} + \Omega_{2} \mu_{2} + a_{1} b_{2} \mu_{1} - a_{1} b_{2} \mu_{2}\right)}{2 a_{1} \mu_{1} \left(- \Omega_{1} + k^{2}\right)} & \frac{\Omega_{1} a_{2} \mu_{1} - \Omega_{2} a_{1} \mu_{2} + a_{1} k^{2} \mu_{1} - a_{2} k^{2} \mu_{2}}{2 a_{1} \mu_{1} \left(- \Omega_{1} + k^{2}\right)} & \frac{k \left(- \Omega_{1} \mu_{1} + \Omega_{2} \mu_{2} - a_{1} b_{2} \mu_{1} + a_{1} b_{2} \mu_{2}\right)}{2 a_{1} \mu_{1} \left(- \Omega_{1} + k^{2}\right)}\\\frac{k \left(- \Omega_{1} \mu_{1} + \Omega_{2} \mu_{2} + a_{2} b_{1} \mu_{1} - a_{2} b_{1} \mu_{2}\right)}{2 b_{1} \mu_{1} \left(- \Omega_{1} + k^{2}\right)} & \frac{- \Omega_{1} b_{2} \mu_{1} - \Omega_{2} b_{1} \mu_{2} + b_{1} k^{2} \mu_{1} + b_{2} k^{2} \mu_{2}}{2 b_{1} \mu_{1} \left(- \Omega_{1} + k^{2}\right)} & \frac{k \left(- \Omega_{1} \mu_{1} + \Omega_{2} \mu_{2} - a_{2} b_{1} \mu_{1} + a_{2} b_{1} \mu_{2}\right)}{2 b_{1} \mu_{1} \left(- \Omega_{1} + k^{2}\right)} & \frac{\Omega_{1} b_{2} \mu_{1} - \Omega_{2} b_{1} \mu_{2} + b_{1} k^{2} \mu_{1} - b_{2} k^{2} \mu_{2}}{2 b_{1} \mu_{1} \left(- \Omega_{1} + k^{2}\right)}\\\frac{\Omega_{1} a_{2} \mu_{1} - \Omega_{2} a_{1} \mu_{2} + a_{1} k^{2} \mu_{1} - a_{2} k^{2} \mu_{2}}{2 a_{1} \mu_{1} \left(- \Omega_{1} + k^{2}\right)} & \frac{k \left(\Omega_{1} \mu_{1} - \Omega_{2} \mu_{2} + a_{1} b_{2} \mu_{1} - a_{1} b_{2} \mu_{2}\right)}{2 a_{1} \mu_{1} \left(- \Omega_{1} + k^{2}\right)} & \frac{- \Omega_{1} a_{2} \mu_{1} - \Omega_{2} a_{1} \mu_{2} + a_{1} k^{2} \mu_{1} + a_{2} k^{2} \mu_{2}}{2 a_{1} \mu_{1} \left(- \Omega_{1} + k^{2}\right)} & \frac{k \left(\Omega_{1} \mu_{1} - \Omega_{2} \mu_{2} - a_{1} b_{2} \mu_{1} + a_{1} b_{2} \mu_{2}\right)}{2 a_{1} \mu_{1} \left(- \Omega_{1} + k^{2}\right)}\\\frac{k \left(\Omega_{1} \mu_{1} - \Omega_{2} \mu_{2} + a_{2} b_{1} \mu_{1} - a_{2} b_{1} \mu_{2}\right)}{2 b_{1} \mu_{1} \left(- \Omega_{1} + k^{2}\right)} & \frac{\Omega_{1} b_{2} \mu_{1} - \Omega_{2} b_{1} \mu_{2} + b_{1} k^{2} \mu_{1} - b_{2} k^{2} \mu_{2}}{2 b_{1} \mu_{1} \left(- \Omega_{1} + k^{2}\right)} & \frac{k \left(\Omega_{1} \mu_{1} - \Omega_{2} \mu_{2} - a_{2} b_{1} \mu_{1} + a_{2} b_{1} \mu_{2}\right)}{2 b_{1} \mu_{1} \left(- \Omega_{1} + k^{2}\right)} & \frac{- \Omega_{1} b_{2} \mu_{1} - \Omega_{2} b_{1} \mu_{2} + b_{1} k^{2} \mu_{1} + b_{2} k^{2} \mu_{2}}{2 b_{1} \mu_{1} \left(- \Omega_{1} + k^{2}\right)}\end{matrix}\right]$

[4]:

Q11 = Q[:2, :2]
Q12 = Q[:2, 2:]
Q21 = Q[2:, :2]
Q22 = Q[2:, 2:]

TD = Q22**(-1)
RD = Q12*TD
RU = -TD*Q21
TU = Q11 - Q12*TD*Q21

TD = sp.together(TD).simplify()
RD = sp.together(RD).simplify()
RU = sp.together(RU).simplify()
TU = sp.together(TU).simplify()

display(Math(latex(RD)))
display(Math(latex(RU)))
display(Math(latex(TD)))
display(Math(latex(TU)))

$\displaystyle \left[\begin{matrix}\frac{k^{2} \left(\Omega_{1} \mu_{1} - \Omega_{2} \mu_{2} + a_{1} b_{2} \mu_{1} - a_{1} b_{2} \mu_{2}\right) \left(\Omega_{1}^{2} \mu_{1} - \Omega_{1} \Omega_{2} \mu_{2} - \Omega_{1} a_{2} b_{1} \mu_{1} + \Omega_{1} a_{2} b_{1} \mu_{2} - \Omega_{1} k^{2} \mu_{1} + \Omega_{2} k^{2} \mu_{2} + a_{2} b_{1} k^{2} \mu_{1} - a_{2} b_{1} k^{2} \mu_{2}\right) - \left(\Omega_{1} a_{2} \mu_{1} - \Omega_{2} a_{1} \mu_{2} + a_{1} k^{2} \mu_{1} - a_{2} k^{2} \mu_{2}\right) \left(\Omega_{1}^{2} b_{2} \mu_{1} + \Omega_{1} \Omega_{2} b_{1} \mu_{2} - \Omega_{1} b_{1} k^{2} \mu_{1} - \Omega_{1} b_{2} k^{2} \mu_{1} - \Omega_{1} b_{2} k^{2} \mu_{2} - \Omega_{2} b_{1} k^{2} \mu_{2} + b_{1} k^{4} \mu_{1} + b_{2} k^{4} \mu_{2}\right)}{\left(\Omega_{1} - k^{2}\right) \left(\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}\right)} & \frac{2 b_{1} k \left(\Omega_{1} \Omega_{2} \mu_{1} \mu_{2} + \Omega_{1} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1} a_{2} b_{2} \mu_{1} \mu_{2} - \Omega_{1} k^{2} \mu_{1}^{2} - \Omega_{2}^{2} \mu_{2}^{2} + \Omega_{2} k^{2} \mu_{1} \mu_{2} - a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + a_{2} b_{2} k^{2} \mu_{2}^{2}\right)}{\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}}\\\frac{2 a_{1} k \left(\Omega_{1} \Omega_{2} \mu_{1} \mu_{2} + \Omega_{1} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1} a_{2} b_{2} \mu_{1} \mu_{2} - \Omega_{1} k^{2} \mu_{1}^{2} - \Omega_{2}^{2} \mu_{2}^{2} + \Omega_{2} k^{2} \mu_{1} \mu_{2} - a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + a_{2} b_{2} k^{2} \mu_{2}^{2}\right)}{\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}} & \frac{k^{2} \left(\Omega_{1} \mu_{1} - \Omega_{2} \mu_{2} + a_{2} b_{1} \mu_{1} - a_{2} b_{1} \mu_{2}\right) \left(\Omega_{1}^{2} \mu_{1} - \Omega_{1} \Omega_{2} \mu_{2} - \Omega_{1} a_{1} b_{2} \mu_{1} + \Omega_{1} a_{1} b_{2} \mu_{2} - \Omega_{1} k^{2} \mu_{1} + \Omega_{2} k^{2} \mu_{2} + a_{1} b_{2} k^{2} \mu_{1} - a_{1} b_{2} k^{2} \mu_{2}\right) - \left(\Omega_{1} b_{2} \mu_{1} - \Omega_{2} b_{1} \mu_{2} + b_{1} k^{2} \mu_{1} - b_{2} k^{2} \mu_{2}\right) \left(\Omega_{1}^{2} a_{2} \mu_{1} + \Omega_{1} \Omega_{2} a_{1} \mu_{2} - \Omega_{1} a_{1} k^{2} \mu_{1} - \Omega_{1} a_{2} k^{2} \mu_{1} - \Omega_{1} a_{2} k^{2} \mu_{2} - \Omega_{2} a_{1} k^{2} \mu_{2} + a_{1} k^{4} \mu_{1} + a_{2} k^{4} \mu_{2}\right)}{\left(\Omega_{1} - k^{2}\right) \left(\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}\right)}\end{matrix}\right]$

$\displaystyle \left[\begin{matrix}\frac{k^{2} \left(\Omega_{1} \mu_{1} - \Omega_{2} \mu_{2} + a_{2} b_{1} \mu_{1} - a_{2} b_{1} \mu_{2}\right) \left(\Omega_{1}^{2} \mu_{1} - \Omega_{1} \Omega_{2} \mu_{2} - \Omega_{1} a_{1} b_{2} \mu_{1} + \Omega_{1} a_{1} b_{2} \mu_{2} - \Omega_{1} k^{2} \mu_{1} + \Omega_{2} k^{2} \mu_{2} + a_{1} b_{2} k^{2} \mu_{1} - a_{1} b_{2} k^{2} \mu_{2}\right) + \left(\Omega_{1} a_{2} \mu_{1} - \Omega_{2} a_{1} \mu_{2} + a_{1} k^{2} \mu_{1} - a_{2} k^{2} \mu_{2}\right) \left(\Omega_{1}^{2} b_{2} \mu_{1} + \Omega_{1} \Omega_{2} b_{1} \mu_{2} - \Omega_{1} b_{1} k^{2} \mu_{1} - \Omega_{1} b_{2} k^{2} \mu_{1} - \Omega_{1} b_{2} k^{2} \mu_{2} - \Omega_{2} b_{1} k^{2} \mu_{2} + b_{1} k^{4} \mu_{1} + b_{2} k^{4} \mu_{2}\right)}{\left(\Omega_{1} - k^{2}\right) \left(\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}\right)} & \frac{2 b_{2} k \left(\Omega_{1}^{2} \mu_{1}^{2} - \Omega_{1} \Omega_{2} \mu_{1} \mu_{2} - \Omega_{1} k^{2} \mu_{1} \mu_{2} + \Omega_{2} a_{1} b_{1} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{1} \mu_{2}^{2} + \Omega_{2} k^{2} \mu_{2}^{2} - a_{1} b_{1} k^{2} \mu_{1}^{2} + a_{1} b_{1} k^{2} \mu_{1} \mu_{2}\right)}{\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}}\\\frac{2 a_{2} k \left(\Omega_{1}^{2} \mu_{1}^{2} - \Omega_{1} \Omega_{2} \mu_{1} \mu_{2} - \Omega_{1} k^{2} \mu_{1} \mu_{2} + \Omega_{2} a_{1} b_{1} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{1} \mu_{2}^{2} + \Omega_{2} k^{2} \mu_{2}^{2} - a_{1} b_{1} k^{2} \mu_{1}^{2} + a_{1} b_{1} k^{2} \mu_{1} \mu_{2}\right)}{\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}} & \frac{k^{2} \left(\Omega_{1} \mu_{1} - \Omega_{2} \mu_{2} + a_{1} b_{2} \mu_{1} - a_{1} b_{2} \mu_{2}\right) \left(\Omega_{1}^{2} \mu_{1} - \Omega_{1} \Omega_{2} \mu_{2} - \Omega_{1} a_{2} b_{1} \mu_{1} + \Omega_{1} a_{2} b_{1} \mu_{2} - \Omega_{1} k^{2} \mu_{1} + \Omega_{2} k^{2} \mu_{2} + a_{2} b_{1} k^{2} \mu_{1} - a_{2} b_{1} k^{2} \mu_{2}\right) + \left(\Omega_{1} b_{2} \mu_{1} - \Omega_{2} b_{1} \mu_{2} + b_{1} k^{2} \mu_{1} - b_{2} k^{2} \mu_{2}\right) \left(\Omega_{1}^{2} a_{2} \mu_{1} + \Omega_{1} \Omega_{2} a_{1} \mu_{2} - \Omega_{1} a_{1} k^{2} \mu_{1} - \Omega_{1} a_{2} k^{2} \mu_{1} - \Omega_{1} a_{2} k^{2} \mu_{2} - \Omega_{2} a_{1} k^{2} \mu_{2} + a_{1} k^{4} \mu_{1} + a_{2} k^{4} \mu_{2}\right)}{\left(\Omega_{1} - k^{2}\right) \left(\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}\right)}\end{matrix}\right]$

$\displaystyle \left[\begin{matrix}\frac{2 a_{1} \mu_{1} \left(\Omega_{1}^{2} b_{2} \mu_{1} + \Omega_{1} \Omega_{2} b_{1} \mu_{2} - \Omega_{1} b_{1} k^{2} \mu_{1} - \Omega_{1} b_{2} k^{2} \mu_{1} - \Omega_{1} b_{2} k^{2} \mu_{2} - \Omega_{2} b_{1} k^{2} \mu_{2} + b_{1} k^{4} \mu_{1} + b_{2} k^{4} \mu_{2}\right)}{\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}} & \frac{2 b_{1} k \mu_{1} \left(\Omega_{1}^{2} \mu_{1} - \Omega_{1} \Omega_{2} \mu_{2} - \Omega_{1} a_{1} b_{2} \mu_{1} + \Omega_{1} a_{1} b_{2} \mu_{2} - \Omega_{1} k^{2} \mu_{1} + \Omega_{2} k^{2} \mu_{2} + a_{1} b_{2} k^{2} \mu_{1} - a_{1} b_{2} k^{2} \mu_{2}\right)}{\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}}\\\frac{2 a_{1} k \mu_{1} \left(\Omega_{1}^{2} \mu_{1} - \Omega_{1} \Omega_{2} \mu_{2} - \Omega_{1} a_{2} b_{1} \mu_{1} + \Omega_{1} a_{2} b_{1} \mu_{2} - \Omega_{1} k^{2} \mu_{1} + \Omega_{2} k^{2} \mu_{2} + a_{2} b_{1} k^{2} \mu_{1} - a_{2} b_{1} k^{2} \mu_{2}\right)}{\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}} & \frac{2 b_{1} \mu_{1} \left(\Omega_{1}^{2} a_{2} \mu_{1} + \Omega_{1} \Omega_{2} a_{1} \mu_{2} - \Omega_{1} a_{1} k^{2} \mu_{1} - \Omega_{1} a_{2} k^{2} \mu_{1} - \Omega_{1} a_{2} k^{2} \mu_{2} - \Omega_{2} a_{1} k^{2} \mu_{2} + a_{1} k^{4} \mu_{1} + a_{2} k^{4} \mu_{2}\right)}{\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}}\end{matrix}\right]$

$\displaystyle \left[\begin{matrix}\frac{2 a_{2} \mu_{2} \left(\Omega_{1} \Omega_{2} b_{2} \mu_{1} - \Omega_{1} b_{2} k^{2} \mu_{1} + \Omega_{2}^{2} b_{1} \mu_{2} - \Omega_{2} b_{1} k^{2} \mu_{1} - \Omega_{2} b_{1} k^{2} \mu_{2} - \Omega_{2} b_{2} k^{2} \mu_{2} + b_{1} k^{4} \mu_{1} + b_{2} k^{4} \mu_{2}\right)}{\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}} & \frac{2 b_{2} k \mu_{2} \left(\Omega_{1} \Omega_{2} \mu_{1} - \Omega_{1} k^{2} \mu_{1} - \Omega_{2}^{2} \mu_{2} - \Omega_{2} a_{2} b_{1} \mu_{1} + \Omega_{2} a_{2} b_{1} \mu_{2} + \Omega_{2} k^{2} \mu_{2} + a_{2} b_{1} k^{2} \mu_{1} - a_{2} b_{1} k^{2} \mu_{2}\right)}{\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}}\\\frac{2 a_{2} k \mu_{2} \left(\Omega_{1} \Omega_{2} \mu_{1} - \Omega_{1} k^{2} \mu_{1} - \Omega_{2}^{2} \mu_{2} - \Omega_{2} a_{1} b_{2} \mu_{1} + \Omega_{2} a_{1} b_{2} \mu_{2} + \Omega_{2} k^{2} \mu_{2} + a_{1} b_{2} k^{2} \mu_{1} - a_{1} b_{2} k^{2} \mu_{2}\right)}{\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}} & \frac{2 b_{2} \mu_{2} \left(\Omega_{1} \Omega_{2} a_{2} \mu_{1} - \Omega_{1} a_{2} k^{2} \mu_{1} + \Omega_{2}^{2} a_{1} \mu_{2} - \Omega_{2} a_{1} k^{2} \mu_{1} - \Omega_{2} a_{1} k^{2} \mu_{2} - \Omega_{2} a_{2} k^{2} \mu_{2} + a_{1} k^{4} \mu_{1} + a_{2} k^{4} \mu_{2}\right)}{\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}}\end{matrix}\right]$

本文档推导方法

[5]:

# [TU, RD],
# [RU, TD],

# left
Pl = sp.Matrix([
    [(-1)*k,            (-1)*b1,           k,            -b2],
    [(-1)*a1,           (-1)*k,            -a2,           k,],
    [(-1)*2*mu1*Omg1,   (-1)*2*k*mu1*b1,   2*mu2*Omg2,     -2*k*mu2*b2,  ],
    [(-1)*2*k*mu1*a1,   (-1)*2*mu1*Omg1,   -2*k*mu2*a2,    2*mu2*Omg2,  ]
])

# right
# (-1) 系数均是由移项形成的
Pr = sp.Matrix([
    [(-1)*k,            (-1)*b2,           k,            -b1],
    [(-1)*a2,           (-1)*k,            -a1,           k,],
    [(-1)*2*mu2*Omg2,   (-1)*2*k*mu2*b2,   2*mu1*Omg1,     -2*k*mu1*b1,  ],
    [(-1)*2*k*mu2*a2,   (-1)*2*mu2*Omg2,   -2*k*mu1*a1,    2*mu1*Omg1,  ]
])

Q = sp.together(sp.expand(Pl.inv() * Pr))

TU2 = Q[:2, :2]
RU2 = Q[2:, :2]
RD2 = Q[:2, 2:]
TD2 = Q[2:, 2:]

display(Math(latex(RD2)))
display(Math(latex(RU2)))
display(Math(latex(TD2)))
display(Math(latex(TU2)))

$\displaystyle \left[\begin{matrix}\frac{- \Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} + \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} - \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} - 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} + 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} + \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} - a_{2} b_{1} k^{4} \mu_{1} \mu_{2} - a_{2} b_{2} k^{4} \mu_{2}^{2}}{\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}} & \frac{2 k \left(\Omega_{1} \Omega_{2} b_{1} \mu_{1} \mu_{2} + \Omega_{1} a_{2} b_{1} b_{2} \mu_{1}^{2} - \Omega_{1} a_{2} b_{1} b_{2} \mu_{1} \mu_{2} - \Omega_{1} b_{1} k^{2} \mu_{1}^{2} - \Omega_{2}^{2} b_{1} \mu_{2}^{2} + \Omega_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} + a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2}\right)}{\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}}\\\frac{2 k \left(\Omega_{1} \Omega_{2} a_{1} \mu_{1} \mu_{2} + \Omega_{1} a_{1} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1} a_{1} a_{2} b_{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} k^{2} \mu_{1}^{2} - \Omega_{2}^{2} a_{1} \mu_{2}^{2} + \Omega_{2} a_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + a_{1} a_{2} b_{2} k^{2} \mu_{2}^{2}\right)}{\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}} & \frac{- \Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} + \Omega_{1}^{2} k^{2} \mu_{1}^{2} - \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} - 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} + 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} + \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} + \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} - a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} - a_{2} b_{2} k^{4} \mu_{2}^{2}}{\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}}\end{matrix}\right]$

$\displaystyle \left[\begin{matrix}\frac{\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} + \Omega_{1}^{2} k^{2} \mu_{1}^{2} - \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} - 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} + \Omega_{2}^{2} k^{2} \mu_{2}^{2} + 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} + \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} - a_{1} b_{1} k^{4} \mu_{1}^{2} - a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}}{\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}} & \frac{2 k \left(\Omega_{1}^{2} b_{2} \mu_{1}^{2} - \Omega_{1} \Omega_{2} b_{2} \mu_{1} \mu_{2} - \Omega_{1} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2} a_{1} b_{1} b_{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{1} b_{2} \mu_{2}^{2} + \Omega_{2} b_{2} k^{2} \mu_{2}^{2} - a_{1} b_{1} b_{2} k^{2} \mu_{1}^{2} + a_{1} b_{1} b_{2} k^{2} \mu_{1} \mu_{2}\right)}{\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}}\\\frac{2 k \left(\Omega_{1}^{2} a_{2} \mu_{1}^{2} - \Omega_{1} \Omega_{2} a_{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2} a_{1} a_{2} b_{1} \mu_{1} \mu_{2} - \Omega_{2} a_{1} a_{2} b_{1} \mu_{2}^{2} + \Omega_{2} a_{2} k^{2} \mu_{2}^{2} - a_{1} a_{2} b_{1} k^{2} \mu_{1}^{2} + a_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2}\right)}{\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}} & \frac{\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} + \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} - \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} - 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} + \Omega_{2}^{2} k^{2} \mu_{2}^{2} + 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} - a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} - a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}}{\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}}\end{matrix}\right]$

$\displaystyle \left[\begin{matrix}\frac{2 \mu_{1} \left(\Omega_{1}^{2} a_{1} b_{2} \mu_{1} + \Omega_{1} \Omega_{2} a_{1} b_{1} \mu_{2} - \Omega_{1} a_{1} b_{1} k^{2} \mu_{1} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{2} - \Omega_{2} a_{1} b_{1} k^{2} \mu_{2} + a_{1} b_{1} k^{4} \mu_{1} + a_{1} b_{2} k^{4} \mu_{2}\right)}{\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}} & \frac{2 k \mu_{1} \left(\Omega_{1}^{2} b_{1} \mu_{1} - \Omega_{1} \Omega_{2} b_{1} \mu_{2} - \Omega_{1} a_{1} b_{1} b_{2} \mu_{1} + \Omega_{1} a_{1} b_{1} b_{2} \mu_{2} - \Omega_{1} b_{1} k^{2} \mu_{1} + \Omega_{2} b_{1} k^{2} \mu_{2} + a_{1} b_{1} b_{2} k^{2} \mu_{1} - a_{1} b_{1} b_{2} k^{2} \mu_{2}\right)}{\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}}\\\frac{2 k \mu_{1} \left(\Omega_{1}^{2} a_{1} \mu_{1} - \Omega_{1} \Omega_{2} a_{1} \mu_{2} - \Omega_{1} a_{1} a_{2} b_{1} \mu_{1} + \Omega_{1} a_{1} a_{2} b_{1} \mu_{2} - \Omega_{1} a_{1} k^{2} \mu_{1} + \Omega_{2} a_{1} k^{2} \mu_{2} + a_{1} a_{2} b_{1} k^{2} \mu_{1} - a_{1} a_{2} b_{1} k^{2} \mu_{2}\right)}{\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}} & \frac{2 \mu_{1} \left(\Omega_{1}^{2} a_{2} b_{1} \mu_{1} + \Omega_{1} \Omega_{2} a_{1} b_{1} \mu_{2} - \Omega_{1} a_{1} b_{1} k^{2} \mu_{1} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{2} - \Omega_{2} a_{1} b_{1} k^{2} \mu_{2} + a_{1} b_{1} k^{4} \mu_{1} + a_{2} b_{1} k^{4} \mu_{2}\right)}{\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}}\end{matrix}\right]$

$\displaystyle \left[\begin{matrix}\frac{2 \mu_{2} \left(\Omega_{1} \Omega_{2} a_{2} b_{2} \mu_{1} - \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} + \Omega_{2}^{2} a_{2} b_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{2} - \Omega_{2} a_{2} b_{2} k^{2} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} + a_{2} b_{2} k^{4} \mu_{2}\right)}{\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}} & \frac{2 k \mu_{2} \left(\Omega_{1} \Omega_{2} b_{2} \mu_{1} - \Omega_{1} b_{2} k^{2} \mu_{1} - \Omega_{2}^{2} b_{2} \mu_{2} - \Omega_{2} a_{2} b_{1} b_{2} \mu_{1} + \Omega_{2} a_{2} b_{1} b_{2} \mu_{2} + \Omega_{2} b_{2} k^{2} \mu_{2} + a_{2} b_{1} b_{2} k^{2} \mu_{1} - a_{2} b_{1} b_{2} k^{2} \mu_{2}\right)}{\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}}\\\frac{2 k \mu_{2} \left(\Omega_{1} \Omega_{2} a_{2} \mu_{1} - \Omega_{1} a_{2} k^{2} \mu_{1} - \Omega_{2}^{2} a_{2} \mu_{2} - \Omega_{2} a_{1} a_{2} b_{2} \mu_{1} + \Omega_{2} a_{1} a_{2} b_{2} \mu_{2} + \Omega_{2} a_{2} k^{2} \mu_{2} + a_{1} a_{2} b_{2} k^{2} \mu_{1} - a_{1} a_{2} b_{2} k^{2} \mu_{2}\right)}{\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}} & \frac{2 \mu_{2} \left(\Omega_{1} \Omega_{2} a_{2} b_{2} \mu_{1} - \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} + \Omega_{2}^{2} a_{1} b_{2} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{2} - \Omega_{2} a_{2} b_{2} k^{2} \mu_{2} + a_{1} b_{2} k^{4} \mu_{1} + a_{2} b_{2} k^{4} \mu_{2}\right)}{\Omega_{1}^{2} a_{2} b_{2} \mu_{1}^{2} - \Omega_{1}^{2} k^{2} \mu_{1}^{2} + \Omega_{1} \Omega_{2} a_{1} b_{2} \mu_{1} \mu_{2} + \Omega_{1} \Omega_{2} a_{2} b_{1} \mu_{1} \mu_{2} + 2 \Omega_{1} \Omega_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{1} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - 2 \Omega_{1} a_{2} b_{2} k^{2} \mu_{1} \mu_{2} + \Omega_{2}^{2} a_{1} b_{1} \mu_{2}^{2} - \Omega_{2}^{2} k^{2} \mu_{2}^{2} - 2 \Omega_{2} a_{1} b_{1} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{1} b_{2} k^{2} \mu_{1} \mu_{2} - \Omega_{2} a_{2} b_{1} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1}^{2} + 2 a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{1} \mu_{2} - a_{1} a_{2} b_{1} b_{2} k^{2} \mu_{2}^{2} + a_{1} b_{1} k^{4} \mu_{1}^{2} + a_{1} b_{2} k^{4} \mu_{1} \mu_{2} + a_{2} b_{1} k^{4} \mu_{1} \mu_{2} + a_{2} b_{2} k^{4} \mu_{2}^{2}}\end{matrix}\right]$

结果对比

直接将两个方法结果的表达式相减，全为0即为通过

[6]:

# RD
test = (RD - RD2).simplify()

test

[6]:

$\displaystyle \left[\begin{matrix}0 & 0\\0 & 0\end{matrix}\right]$

[7]:

# RU
test = (RU - RU2).simplify()

test

[7]:

$\displaystyle \left[\begin{matrix}0 & 0\\0 & 0\end{matrix}\right]$

[8]:

# TD
test = (TD - TD2).simplify()

test

[8]:

$\displaystyle \left[\begin{matrix}0 & 0\\0 & 0\end{matrix}\right]$

[9]:

# TU
test = (TU - TU2).simplify()

test

[9]:

$\displaystyle \left[\begin{matrix}0 & 0\\0 & 0\end{matrix}\right]$