From 9a21df950004eb97cc9b1bc8d32a39e57e2549ac Mon Sep 17 00:00:00 2001 From: Dengda98 Date: Fri, 20 Jun 2025 17:13:07 +0800 Subject: [PATCH] add RT.rst and RT_formula.ipynb --- docs/source/Formula/RT.rst | 274 ++++++++++++++++++ docs/source/Formula/RT_formula.ipynb | 402 +++++++++++++++++++++++++++ docs/source/Formula/index.rst | 1 + 3 files changed, 677 insertions(+) create mode 100644 docs/source/Formula/RT.rst create mode 100644 docs/source/Formula/RT_formula.ipynb diff --git a/docs/source/Formula/RT.rst b/docs/source/Formula/RT.rst new file mode 100644 index 00000000..201a8c78 --- /dev/null +++ b/docs/source/Formula/RT.rst @@ -0,0 +1,274 @@ +反射透射系数矩阵的另一种推导 +============================== + +:Author: Zhu Dengda +:Email: zhudengda@mail.iggcas.ac.cn + +----------------------------------------------------------- + +在 :ref:`《理论地震图及其应用（初稿）》 ` 中给出了反射透射系数矩阵（下称 R/T 矩阵）的推导，其中推导的基础是先计算出固体-固体界面两侧的势函数转换矩阵 :math:`\mathbf{Q}(j^-,j^+) = \mathbf{D}_{j-1}^{-1} \mathbf{D}_{j}` ，但这种推导不太适用于后续对于液体-固体界面的 R/T 矩阵的推导，故这里给出另一种推导方式，本质还是矩阵的各种变换。 + +原推导过程涉及多种矢量符号，比较容易产生混淆，以下尽量将矩阵、矢量展开。 + +以下推导以 P-SV 情况为例。 + +最终每一项系数的详细表达式我使用 Python 库 `SymPy `_ 辅助推导，包括与原推导结果的对比验证，可在这里下载 :download:`RT_formula.ipynb` （ :doc:`预览 ` ）。 + +---------------------------------------- + + +界面两侧的势函数关系 +----------------------------- + +从波动方程出发，使用柱面谐矢量(vector cylindrical harmonics)表示位移和应力，则某层内的位移势函数与位移应力矢量之间的关系可以表示为（略去角标 :math:`j`） + +.. math:: + :label: + + \begin{bmatrix} + q_m \\ + w_m \\ + \sigma_{Rm} \\ + \tau_{Rm} \\ + \end{bmatrix} = + \begin{bmatrix} + k & b & k & -b \\ + a & k & -a & k \\ + 2\mu\Omega & 2k\mu b & 2\mu\Omega & -2k\mu b \\ + 2k\mu a & 2\mu\Omega & -2k\mu a & 2\mu\Omega \\ + \end{bmatrix} + \begin{bmatrix} + \phi_m^- \\ + \psi_m^- \\ + \phi_m^+ \\ + \psi_m^+ \\ + \end{bmatrix} + +其中左侧为位移应力矢量，右侧的 4x4 矩阵 我们称为 :math:`D` 矩阵。为简单起见，我们假设界面位于第1/2层之间，根据应力位移连续的边界条件，有 + +.. math:: + :label: layer12 + + & \begin{bmatrix} + k & b_1 & k & -b_1 \\ + a_1 & k & -a_1 & k \\ + 2\mu_1\Omega_1 & 2k\mu_1 b_1 & 2\mu_1\Omega_1 & -2k\mu_1 b_1 \\ + 2k\mu_1 a_1 & 2\mu_1\Omega_1 & -2k\mu_1 a_1 & 2\mu_1\Omega_1 \\ + \end{bmatrix} + \begin{bmatrix} + \phi_m^- (z_1^-) \\ + \psi_m^- (z_1^-) \\ + \phi_m^+ (z_1^-) \\ + \psi_m^+ (z_1^-) \\ + \end{bmatrix} \\ + = & \begin{bmatrix} + k & b_2 & k & -b_2 \\ + a_2 & k & -a_2 & k \\ + 2\mu_2\Omega_2 & 2k\mu_2 b_2 & 2\mu_2\Omega_2 & -2k\mu_2 b_2 \\ + 2k\mu_2 a_2 & 2\mu_2\Omega_2 & -2k\mu_2 a_2 & 2\mu_2\Omega_2 \\ + \end{bmatrix} + \begin{bmatrix} + \phi_m^- (z_1^+) \\ + \psi_m^- (z_1^+) \\ + \phi_m^+ (z_1^+) \\ + \psi_m^+ (z_1^+) \\ + \end{bmatrix} + +界面上的 R/T 矩阵 +-------------------------- + +原推导中对 :eq:`layer12` 式做了矩阵求逆以获得两侧势函数的转换矩阵。这里我们先保留该形式，仍然根据入射方向分两种情况讨论，最终同样可求出 R/T 矩阵。 + +波从上向下入射 +~~~~~~~~~~~~~~~~ + +此时下层没有向上传播的入射波，即 :math:`[\phi_m^- (z_1^+), \psi_m^- (z_1^+)]^T = \mathbf{0}` ，:eq:`layer12` 式变为 + +.. math:: + :label: + + \begin{bmatrix} + k & b_1 & k & -b_1 \\ + a_1 & k & -a_1 & k \\ + 2\mu_1\Omega_1 & 2k\mu_1 b_1 & 2\mu_1\Omega_1 & -2k\mu_1 b_1 \\ + 2k\mu_1 a_1 & 2\mu_1\Omega_1 & -2k\mu_1 a_1 & 2\mu_1\Omega_1 \\ + \end{bmatrix} + \begin{bmatrix} + \phi_m^- (z_1^-) \\ + \psi_m^- (z_1^-) \\ + \bbox[yellow] {\phi_m^+ (z_1^-)} \\ + \bbox[yellow] {\psi_m^+ (z_1^-)} \\ + \end{bmatrix} = \begin{bmatrix} + k & -b_2 \\ + -a_2 & k \\ + 2\mu_2\Omega_2 & -2k\mu_2 b_2 \\ + -2k\mu_2 a_2 & 2\mu_2\Omega_2 \\ + \end{bmatrix} + \begin{bmatrix} + \phi_m^+ (z_1^+) \\ + \psi_m^+ (z_1^+) \\ + \end{bmatrix} + +其中高亮部分的势函数为当前情况的“已知项”，通过移项+矩阵重排的方式可得到 + +.. math:: + :label: U2D + + \begin{bmatrix} + -k & -b_1 & k & -b_2 \\ + -a_1 & -k & -a_2 & k \\ + -2\mu_1\Omega_1 & -2k\mu_1 b_1 & 2\mu_2\Omega_2 & -2k\mu_2 b_2 \\ + -2k\mu_1 a_1 & -2\mu_1\Omega_1 & -2k\mu_2 a_2 & 2\mu_2\Omega_2 \\ + \end{bmatrix} + \begin{bmatrix} + \phi_m^- (z_1^-) \\ + \psi_m^- (z_1^-) \\ + \phi_m^+ (z_1^+) \\ + \psi_m^+ (z_1^+) \\ + \end{bmatrix} = + \begin{bmatrix} + k & -b_1 \\ + -a_1 & k \\ + 2\mu_1\Omega_1 & -2k\mu_1 b_1 \\ + -2k\mu_1 a_1 & 2\mu_1\Omega_1 \\ + \end{bmatrix} + \begin{bmatrix} + \bbox[yellow] {\phi_m^+ (z_1^-)} \\ + \bbox[yellow] {\psi_m^+ (z_1^-)} \\ + \end{bmatrix} + +其中等号左边矩阵前两列的负号由移项产生，此时左边的势函数矢量（作为未知量）已经变成两层的混合版本，适定方程可简单使用逆矩阵求解，得到 + +.. math:: + :label: + + \begin{bmatrix} + \phi_m^- (z_1^-) \\ + \psi_m^- (z_1^-) \\ + \end{bmatrix} = + \mathbf{R}_D^{2\times2} + \begin{bmatrix} + \bbox[yellow] {\phi_m^+ (z_1^-)} \\ + \bbox[yellow] {\psi_m^+ (z_1^-)} \\ + \end{bmatrix} + + \begin{bmatrix} + \phi_m^+ (z_1^+) \\ + \psi_m^+ (z_1^+) \\ + \end{bmatrix} = + \mathbf{T}_D^{2\times2} + \begin{bmatrix} + \bbox[yellow] {\phi_m^+ (z_1^-)} \\ + \bbox[yellow] {\psi_m^+ (z_1^-)} \\ + \end{bmatrix} + +波从下向上入射 +~~~~~~~~~~~~~~~~ + +此时上层没有向下传播的入射波，即 :math:`[\phi_m^+ (z_1^-), \psi_m^+ (z_1^-)]^T = \mathbf{0}` ，:eq:`layer12` 式变为 + +.. math:: + :label: + + \begin{bmatrix} + k & b_1 \\ + a_1 & k \\ + 2\mu_1\Omega_1 & 2k\mu_1 b_1 \\ + 2k\mu_1 a_1 & 2\mu_1\Omega_1 \\ + \end{bmatrix} + \begin{bmatrix} + \phi_m^- (z_1^-) \\ + \psi_m^- (z_1^-) \\ + \end{bmatrix} = \begin{bmatrix} + k & b_2 & k & -b_2 \\ + a_2 & k & -a_2 & k \\ + 2\mu_2\Omega_2 & 2k\mu_2 b_2 & 2\mu_2\Omega_2 & -2k\mu_2 b_2 \\ + 2k\mu_2 a_2 & 2\mu_2\Omega_2 & -2k\mu_2 a_2 & 2\mu_2\Omega_2 \\ + \end{bmatrix} + \begin{bmatrix} + \bbox[yellow] {\phi_m^- (z_1^+)} \\ + \bbox[yellow] {\psi_m^- (z_1^+)} \\ + \phi_m^+ (z_1^+) \\ + \psi_m^+ (z_1^+) \\ + \end{bmatrix} + +其中高亮部分的势函数同样为当前情况的“已知项”，为保持与 :eq:`U2D` 式的形式匹配，通过类似的移项+矩阵重排的方式可得到 + +.. math:: + :label: D2U + + \begin{bmatrix} + -k & -b_1 & k & -b_2 \\ + -a_1 & -k & -a_2 & k \\ + -2\mu_1\Omega_1 & -2k\mu_1 b_1 & 2\mu_2\Omega_2 & -2k\mu_2 b_2 \\ + -2k\mu_1 a_1 & -2\mu_1\Omega_1 & -2k\mu_2 a_2 & 2\mu_2\Omega_2 \\ + \end{bmatrix} + \begin{bmatrix} + \phi_m^- (z_1^-) \\ + \psi_m^- (z_1^-) \\ + \phi_m^+ (z_1^+) \\ + \psi_m^+ (z_1^+) \\ + \end{bmatrix} = + \begin{bmatrix} + -k & -b_2 \\ + -a_2 & -k \\ + -2\mu_2\Omega_2 & -2k\mu_2 b_2 \\ + -2k\mu_2 a_2 & -2\mu_2\Omega_2 \\ + \end{bmatrix} + \begin{bmatrix} + \bbox[yellow] {\phi_m^- (z_1^+)} \\ + \bbox[yellow] {\psi_m^- (z_1^+)} \\ + \end{bmatrix} + +矩阵中的负号由移项产生，等号左边形式与 :eq:`U2D` 式完全一致。同样该适定方程可简单使用逆矩阵求解，得到 + +.. math:: + :label: + + \begin{bmatrix} + \phi_m^- (z_1^-) \\ + \psi_m^- (z_1^-) \\ + \end{bmatrix} = + \mathbf{T}_U^{2\times2} + \begin{bmatrix} + \bbox[yellow] {\phi_m^- (z_1^+)} \\ + \bbox[yellow] {\psi_m^- (z_1^+)} \\ + \end{bmatrix} + + \begin{bmatrix} + \phi_m^+ (z_1^+) \\ + \psi_m^+ (z_1^+) \\ + \end{bmatrix} = + \mathbf{R}_U^{2\times2} + \begin{bmatrix} + \bbox[yellow] {\phi_m^- (z_1^+)} \\ + \bbox[yellow] {\psi_m^- (z_1^+)} \\ + \end{bmatrix} + +合并求解 +~~~~~~~~~~ + +:eq:`U2D` 式和 :eq:`D2U` 式可合并，一并使用逆矩阵求得最终界面上的 R/T 矩阵， + +.. math:: + :label: + + & \begin{bmatrix} + \mathbf{T}_U^{2\times2} & \mathbf{R}_D^{2\times2} \\ + \mathbf{R}_U^{2\times2} & \mathbf{T}_D^{2\times2} \\ + \end{bmatrix} \\ + = & + \begin{bmatrix} + -k & -b_1 & k & -b_2 \\ + -a_1 & -k & -a_2 & k \\ + -2\mu_1\Omega_1 & -2k\mu_1 b_1 & 2\mu_2\Omega_2 & -2k\mu_2 b_2 \\ + -2k\mu_1 a_1 & -2\mu_1\Omega_1 & -2k\mu_2 a_2 & 2\mu_2\Omega_2 \\ + \end{bmatrix}^{-1} + \begin{bmatrix} + -k & -b_2 & k & -b_1 \\ + -a_2 & -k & -a_1 & k \\ + -2\mu_2\Omega_2 & -2k\mu_2 b_2 & 2\mu_1\Omega_1 & -2k\mu_1 b_1 \\ + -2k\mu_2 a_2 & -2\mu_2\Omega_2 & -2k\mu_1 a_1 & 2\mu_1\Omega_1 \\ + \end{bmatrix} + +之后的操作如增加时间延迟因子，广义 R/T 矩阵递推等不受影响。 \ No newline at end of file diff --git a/docs/source/Formula/RT_formula.ipynb b/docs/source/Formula/RT_formula.ipynb new file mode 100644 index 00000000..030f5795 --- /dev/null +++ b/docs/source/Formula/RT_formula.ipynb @@ -0,0 +1,402 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# 反射透射系数详细表达式\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 +} diff --git a/docs/source/Formula/index.rst b/docs/source/Formula/index.rst index 9f033b4a..177550d6 100644 --- a/docs/source/Formula/index.rst +++ b/docs/source/Formula/index.rst @@ -19,6 +19,7 @@ static_uniform static_source + RT