{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# 反射透射系数矩阵详细表达式\n", "\n", "+ Author: Zhu Dengda \n", "+ Email: zhudengda@mail.iggcas.ac.cn" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "np.set_printoptions(linewidth=200)\n", "\n", "import sympy as sp\n", "from sympy.printing.latex import latex\n", "from IPython.display import display, Math" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "# 定义基本变量\n", "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')\n", "b2, a2, mu2, Omg2, kb2, ka2 = sp.symbols(r'b_2 a_2 mu_2 \\Omega_2 k_{b2} k_{a2}')\n", "lam1, lam2 = sp.symbols(r'\\lambda_1 \\lambda_2')\n", "rho1, rho2 = sp.symbols(r'\\rho_1 \\rho_2')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 原推导方法" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\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]$" ], "text/plain": [ "Matrix([\n", "[(-\\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*(-\\Omega_1 + k**2)), k*(-\\Omega_1*mu_1 + \\Omega_2*mu_2 + a_1*b_2*mu_1 - a_1*b_2*mu_2)/(2*a_1*mu_1*(-\\Omega_1 + k**2)), (\\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*(-\\Omega_1 + k**2)), k*(-\\Omega_1*mu_1 + \\Omega_2*mu_2 - a_1*b_2*mu_1 + a_1*b_2*mu_2)/(2*a_1*mu_1*(-\\Omega_1 + k**2))],\n", "[ k*(-\\Omega_1*mu_1 + \\Omega_2*mu_2 + a_2*b_1*mu_1 - a_2*b_1*mu_2)/(2*b_1*mu_1*(-\\Omega_1 + k**2)), (-\\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*(-\\Omega_1 + k**2)), k*(-\\Omega_1*mu_1 + \\Omega_2*mu_2 - a_2*b_1*mu_1 + a_2*b_1*mu_2)/(2*b_1*mu_1*(-\\Omega_1 + k**2)), (\\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*(-\\Omega_1 + k**2))],\n", "[ (\\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*(-\\Omega_1 + k**2)), k*(\\Omega_1*mu_1 - \\Omega_2*mu_2 + a_1*b_2*mu_1 - a_1*b_2*mu_2)/(2*a_1*mu_1*(-\\Omega_1 + k**2)), (-\\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*(-\\Omega_1 + k**2)), k*(\\Omega_1*mu_1 - \\Omega_2*mu_2 - a_1*b_2*mu_1 + a_1*b_2*mu_2)/(2*a_1*mu_1*(-\\Omega_1 + k**2))],\n", "[ k*(\\Omega_1*mu_1 - \\Omega_2*mu_2 + a_2*b_1*mu_1 - a_2*b_1*mu_2)/(2*b_1*mu_1*(-\\Omega_1 + k**2)), (\\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*(-\\Omega_1 + k**2)), k*(\\Omega_1*mu_1 - \\Omega_2*mu_2 - a_2*b_1*mu_1 + a_2*b_1*mu_2)/(2*b_1*mu_1*(-\\Omega_1 + k**2)), (-\\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*(-\\Omega_1 + k**2))]])" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ds1 = sp.Matrix([\n", " [k, b1, k, -b1],\n", " [a1, k, -a1, k],\n", " [2*mu1*Omg1, 2*k*mu1*b1, 2*mu1*Omg1, -2*k*mu1*b1],\n", " [2*k*mu1*a1, 2*mu1*Omg1, -2*k*mu1*a1, 2*mu1*Omg1]\n", "])\n", "\n", "Ds2 = sp.Matrix([\n", " [k, b2, k, -b2],\n", " [a2, k, -a2, k],\n", " [2*mu2*Omg2, 2*k*mu2*b2, 2*mu2*Omg2, -2*k*mu2*b2],\n", " [2*k*mu2*a2, 2*mu2*Omg2, -2*k*mu2*a2, 2*mu2*Omg2]\n", "])\n", "\n", "Q = sp.together(sp.expand(Ds1.inv() * Ds2))\n", "Q" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\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]$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\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]$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\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]$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\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]$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "Q11 = Q[:2, :2]\n", "Q12 = Q[:2, 2:]\n", "Q21 = Q[2:, :2]\n", "Q22 = Q[2:, 2:]\n", "\n", "TD = Q22**(-1)\n", "RD = Q12*TD\n", "RU = -TD*Q21\n", "TU = Q11 - Q12*TD*Q21\n", "\n", "TD = sp.together(TD).simplify()\n", "RD = sp.together(RD).simplify()\n", "RU = sp.together(RU).simplify()\n", "TU = sp.together(TU).simplify()\n", "\n", "display(Math(latex(RD)))\n", "display(Math(latex(RU)))\n", "display(Math(latex(TD)))\n", "display(Math(latex(TU)))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 本文档推导方法" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\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]$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\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]$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\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]$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\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]$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# [TU, RD],\n", "# [RU, TD],\n", "\n", "# left\n", "Pl = sp.Matrix([\n", " [(-1)*k, (-1)*b1, k, -b2],\n", " [(-1)*a1, (-1)*k, -a2, k,],\n", " [(-1)*2*mu1*Omg1, (-1)*2*k*mu1*b1, 2*mu2*Omg2, -2*k*mu2*b2, ],\n", " [(-1)*2*k*mu1*a1, (-1)*2*mu1*Omg1, -2*k*mu2*a2, 2*mu2*Omg2, ]\n", "])\n", "\n", "# right\n", "# (-1) 系数均是由移项形成的\n", "Pr = sp.Matrix([\n", " [(-1)*k, (-1)*b2, k, -b1],\n", " [(-1)*a2, (-1)*k, -a1, k,],\n", " [(-1)*2*mu2*Omg2, (-1)*2*k*mu2*b2, 2*mu1*Omg1, -2*k*mu1*b1, ],\n", " [(-1)*2*k*mu2*a2, (-1)*2*mu2*Omg2, -2*k*mu1*a1, 2*mu1*Omg1, ]\n", "])\n", "\n", "Q = sp.together(sp.expand(Pl.inv() * Pr))\n", "\n", "TU2 = Q[:2, :2]\n", "RU2 = Q[2:, :2]\n", "RD2 = Q[:2, 2:]\n", "TD2 = Q[2:, 2:]\n", "\n", "display(Math(latex(RD2)))\n", "display(Math(latex(RU2)))\n", "display(Math(latex(TD2)))\n", "display(Math(latex(TU2)))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 结果对比\n", "直接将两个方法结果的表达式相减，全为0即为通过" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & 0\\\\0 & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[0, 0],\n", "[0, 0]])" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# RD\n", "test = (RD - RD2).simplify()\n", "\n", "test" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & 0\\\\0 & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[0, 0],\n", "[0, 0]])" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# RU\n", "test = (RU - RU2).simplify()\n", "\n", "test" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & 0\\\\0 & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[0, 0],\n", "[0, 0]])" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# TD\n", "test = (TD - TD2).simplify()\n", "\n", "test" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}0 & 0\\\\0 & 0\\end{matrix}\\right]$" ], "text/plain": [ "Matrix([\n", "[0, 0],\n", "[0, 0]])" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# TU\n", "test = (TU - TU2).simplify()\n", "\n", "test" ] } ], "metadata": { "kernelspec": { "display_name": "py310", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.10.14" } }, "nbformat": 4, "nbformat_minor": 2 }