machinelearning_notebook/1_numpy_matplotlib_scipy_sympy/5-sympy_tutorial.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Sympy - Python中的符号代数"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "J.R. Johansson (jrjohansson at gmail.com)\n",
    "\n",
    "这门课的最新版本[IPython notebook](http://ipython.org/notebook.html) 可以在这里获得[http://github.com/jrjohansson/scientific-python-lectures](http://github.com/jrjohansson/scientific-python-lectures).\n",
    "\n",
    "这节课其他的参考书可以在这里找到[http://jrjohansson.github.io](http://jrjohansson.github.io) 。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. 简介"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "有两个值得注意的Python计算机代数系统（CAS）:\n",
    "\n",
    "* [SymPy](http://sympy.org/en/index.html) - 这是一个python模块可以被用在任意的Python程序或者是IPython会话中，它提供了强大的计算机代数系统（CAS）特征。\n",
    "* [Sage](http://www.sagemath.org/) - Sage 是一个全功能的且非常强大的CAS环境，它的目的是提供一个与Mathematica和Maple竞争的开源系统。Sage并不是一个常规的Python模块，而是一个CAS环境用Python作为其编程语言。\n",
    "\n",
    "Sage在某些方面比SymPy更加强大，但是他们都提供非常完善的CAS功能。SymPy的优点是，它是一个常规的Python模块而且和IPython notebook集成的很好。\n",
    "\n",
    "在这个课上我们将会学会如何运用SymPy和IPython notebooks。如果你对CAS开源环境感兴趣，我也推荐你去阅读更多关于Sage的内容。\n",
    "\n",
    "为了开始在Python程序或笔记本中使用SymPy，首先导入模块`sympy`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "为了得到一个看起来更好的$\\LaTeX$格式输出运行："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "init_printing()\n",
    "\n",
    "# 或者使用较旧版本的sympy/ipython，加载ipython扩展\n",
    "#%load_ext sympy.interactive.ipythonprinting\n",
    "# 或者\n",
    "#%load_ext sympyprinting"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. 符号变量"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "在SymPy中我们需要为需要处理的变量创造符号。我们可以创建一个新的符号用`Symbol`类："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = Symbol('x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(x + \\pi\\right)^{2}$"
      ],
      "text/plain": [
       "       2\n",
       "(x + π) "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(pi + x)**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 定义符号的可替代的方式\n",
    "a, b, c = symbols(\"a, b, c\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sympy.core.symbol.Symbol"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们可以给符号加上假设当我们创造他们的时候："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = Symbol('x', real=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x.is_imaginary"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = Symbol('x', positive=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\text{True}$"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x > 0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.1 复数"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "虚数单位在Sympy中被记做`I`。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 1 + i$"
      ],
      "text/plain": [
       "1 + ⅈ"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1+1*I"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABgAAAAOCAYAAAA1+Nx+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAiElEQVQ4EWP8//8/A7VAQ0ODEdCs1UBsDGR/AJnLAiIoAUCDBID6ZwPxOyA2AWIlIIYDalgAcmkoyESgZWVACuQLOGCCs2jEGLWAYMDSPIhYoMlsL9ApoORGLAgF6jtHjGKQBaBkZkyMYnLU0DyIhr4FjNQo7IDxCCrgQIkEVBaB6HtADEoEpwEsISASVdEGngAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle -1$"
      ],
      "text/plain": [
       "-1"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(i x + 1\\right)^{2}$"
      ],
      "text/plain": [
       "         2\n",
       "(ⅈ⋅x + 1) "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(x * I + 1)**2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.2 有理数"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "在SymPy中有三个不同的数学表达形式`Real`, `Rational`, `Integer`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "r1 = Rational(4,5)\n",
    "r2 = Rational(5,4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{4}{5}$"
      ],
      "text/plain": [
       "4/5"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{41}{20}$"
      ],
      "text/plain": [
       "41\n",
       "──\n",
       "20"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r1+r2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{16}{25}$"
      ],
      "text/plain": [
       "16\n",
       "──\n",
       "25"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r1/r2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3 数值评估"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "SymPy使用任意精度的库作为数值后端，而且预定义了许多数学常数，例如：`pi`, `e`,和用`oo`代表无限。\n",
    "\n",
    "为了数值地衡量一个表达，我们可以用`evalf`方程（或者`N`）。它使用了一个参数`n`确定了有效数字的数量。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 3.1415926535897932384626433832795028841971693993751$"
      ],
      "text/plain": [
       "3.1415926535897932384626433832795028841971693993751"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pi.evalf(n=50)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "y = (x + pi)**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 9.8696 \\left(0.31831 x + 1\\right)^{2}$"
      ],
      "text/plain": [
       "                      2\n",
       "9.8696⋅(0.31831⋅x + 1) "
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N(y, 5) # 和evalf 一样"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "当我们数值地衡量代数表达式，我们通常希望带入一个具有数值的符号。在SymPy中我们用`subs`方程来这样做："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(1.5 + \\pi\\right)^{2}$"
      ],
      "text/plain": [
       "         2\n",
       "(1.5 + π) "
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y.subs(x, 1.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 21.5443823618587$"
      ],
      "text/plain": [
       "21.5443823618587"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N(y.subs(x, 1.5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`subs`当然也可以被用来代入符号和表达："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(a + 2 \\pi\\right)^{2}$"
      ],
      "text/plain": [
       "         2\n",
       "(a + 2⋅π) "
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y.subs(x, a+pi)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们也可以将表达式的数值演化和NumPy数组结合："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "x_vec = numpy.arange(0, 10, 0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "y_vec = numpy.array([N(((x + pi)**2).subs(x, xx)) for xx in x_vec])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "ax.plot(x_vec, y_vec);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "然而，这个类型的数值演化会是非常慢的，有一种非常有效的方式去做：使用函数`lambdify`去“编译”一个SymPy表达并代入函数再进行数值计算是非常高效的。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = lambdify([x], (x + pi)**2, 'numpy')  # 第一个参数是一个变量列表\n",
    "                                         # f将是以下函数的函数:本例中仅为x -> f(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "y_vec = f(x_vec)  # 现在我们可以之间传入一个numpy数组而f(x)会被更有效地计算"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "在使用“speedup”函数而不是直接的进行数值计算时的加速是非常重要的，通常相差几个数量级。即使实在简单的例子中我们也会得到一个明显的加速："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "10 loops, best of 3: 28.2 ms per loop\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "\n",
    "y_vec = numpy.array([N(((x + pi)**2).subs(x, xx)) for xx in x_vec])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The slowest run took 8.86 times longer than the fastest. This could mean that an intermediate result is being cached \n",
      "100000 loops, best of 3: 2.93 µs per loop\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "\n",
    "y_vec = f(x_vec)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4. 代数操作"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "CAS的一个主要用处是执行表达的代数操作。例如，我们可能想要展开一个乘积，一个表达式的因式，或者只是一个表达式。在本节中演示了在SymPy中进行这些基本操作的功能。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 4.1 展开和乘积"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "代数操作中的第一步："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(x + 1\\right) \\left(x + 2\\right) \\left(x + 3\\right)$"
      ],
      "text/plain": [
       "(x + 1)⋅(x + 2)⋅(x + 3)"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(x+1)*(x+2)*(x+3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKsAAAAVCAYAAAA5KB4cAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAEOUlEQVRoBe2a61EbMRDHDeMCSNIB6QBIBUAHkFQAdBCGb/7GQAcJFRDogKQCHiWQCuJxB87/d5HOyvlsn3S6J96ZtXR7euxL0q7OG9PpdOALo9FoW32OhBPhlvCT8FL0F5W9AMmCXBdGGOQFTkRH5jU0oIFh4JxX6vcqw13TX+VXFb+E73juCVxJrjMri+rfVH8WfrS0dVmvBjYDp7tUv1un7wfVX53nPlRP5aAHjiAs0G3RdhzaulqjBoJ2Vhkse9wTEmDMPgG76lOfBOq6LBshMasVWk7L8f9FeKt6EhLYd30rJR+L8UjlOgxoyLilnBWeZTwSkTvhg+qNOqzmz+7uLKLsKQDbXqAxOPqRcVf1xhKsonwUbeelhBobi/9cOwaFAS7fGE94Ltqzyp/C0s7hjl+krjnJ1h+EZ/BAH5U4F1hqJzRjo7xGHFXzsxncCMfCPaG9mVB1BkXbzXq0r2Z0vdCO3s5qlPJbou6rbh3T7jYo09IKa0PjkMiQvHwv3On/hjglCyVxVPMKI5dK+jQejsGtwCFjqmSHZXF6j6s+QTKqH7o9NvMTduUmeEXbMU6VECqn4WmpHUOcFWOxyl2DYQiU+sNM6lvgWKA3iBeSOwy473YWPXEwl+ZTV38cleuqc9Wtg5B0cYqEQLCMIZM12CdIziJ29HZWowRW+oUm+GOe+SjQyDEJH0IW0MTwEqvgThXFU6agedK715S4rsTQwEo7ps4qI2CYUzMrznciZHch2wce1eaeikqOeu/jnr4VALvei3iihFcWEHHqnWhuWADfPjL26QOH1DEDHz3MelVeW2nHTYcFYrNrULRHIUH9gZ459jByNkMTqVkwSrdM7MEr/AvZ/XBWQgQXOiejy3zEeqv0IDvhXxYW2jFxVjVmR806I4ZOdlKV73Pei9Qa2JEM2eSM+PnGKqIHMkZRdsv1sNSONgx4khBuwkRywtGa0FQm2WhZbWkcEhaSsSywGAZ6nxcPwkfu/KITq9LV5Z1ngFiTRcgNBeFAJ2UU37Ehih6k92i2LGrHxFnVOBt/4lChWe9C5WqePGcciM4uztVVyEcFHHWycFJzL6mxuyzjEvH8XsXSg8aJbcuVdnRj1kRqMWF3PhsCDETbAv3UUltrhFzG29yu20EZK1Fmy/Sw0o5D44RcxhJ0c1wmK0Z118hcU0XfaSNZgFibrx5Z2BWBMIGPBThzl2XMyhb03HI9rLQjOys7KTg2woxVT49V0XjH7UArQfyxwHDINEE0cnwWnes3oNMy/hMhym9r9VDEjkOpAGOTSSPIQJ34vs6fUgigSVJw4jQkoE3bQPwdCjkZ4BkgYXM/B3daRsnFqcDpQLII8D8MYnDuvtM4v0C7VutB/C+1Y+l/XSWqK/kjJsskWCVnr6f7W5ARTVYp51yCVY/p5mYh7EhDj7m3/SC8BRmxVGVy/gUdDUuoVh6pGQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle x^{3} + 6 x^{2} + 11 x + 6$"
      ],
      "text/plain": [
       " 3      2           \n",
       "x  + 6⋅x  + 11⋅x + 6"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand((x+1)*(x+2)*(x+3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`expand`函数有一些关键字参数，我们可以告诉函数我们想要执行什么样的扩展。例如，为了扩展三角函数表达式，用`trig=True`的关键字参数："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\sin{\\left(a + b \\right)}$"
      ],
      "text/plain": [
       "sin(a + b)"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sin(a+b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\sin{\\left(a \\right)} \\cos{\\left(b \\right)} + \\sin{\\left(b \\right)} \\cos{\\left(a \\right)}$"
      ],
      "text/plain": [
       "sin(a)⋅cos(b) + sin(b)⋅cos(a)"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(sin(a+b), trig=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "有关`expand`函数可以执行的各种扩展类型的详细说明，请参见`help(expand)`。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "与乘积展开相反的当然就是因式分解了。在SymPy中用`factor`函数来进行因式分解表达："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(x + 1\\right) \\left(x + 2\\right) \\left(x + 3\\right)$"
      ],
      "text/plain": [
       "(x + 1)⋅(x + 2)⋅(x + 3)"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(x**3 + 6 * x**2 + 11*x + 6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 4.2 化简"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`simplify`尝试使用不同的技巧，去将表达简化到易懂的形式。也有很多可以替代`simplify`的函数：`trigsimp`, `powsimp`, `logcombine`等等。\n",
    "\n",
    "这些函数的基本用法如下："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMIAAAAVCAYAAAAU2jGmAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAE/0lEQVRoBeWb71XbMBTF3R4GgI5ANwgwQWEDoBMUNqCHT/CVblA6AYUNQicAsgGMQLMBvT9F8lFs2SgJpH7OO+ehP5Yl3Ss9PUkxxcvLSxHr2dnZZpy2Gn8LHG9RR1f4WxTLou93hQf6kcLysYjk/Pz8RMlBlGU5uunxzIWhZ1zAwdx8rAIXH7AQRGD3Fewo/O4yDPxRXzHaa+mW4uNql5WHYT8pvKk+a0urvCku1N9N4Qnjtq34M2nlj2KcSs/Mh1Eujj3udYVwcyEct21cOENQIV74o3ArLtzFuO/rL/WNwWbQMYYN5dcMQfmF8h8UfGl6TplYVM4MF/Rb/Q0DfRBwKO9CcSb9nuLVCZDNh961xgX9ZdIHQygUP1LeT+mB4lMLotIlF2FrBHEUXpqoE7u+kzO1qXfGUkAB9irjZXCBL1fMcOEB0d9vMThxg3dgYcBbVmUWPqxxwaQ/En48epDfPnIaMqKw5CIYwqFevowKLCOK9aLvKh4X+HLbssbFrgj8m8CHJ1hXPh6jlBn5sMYFW0EWANSJ8JbxkBfCmIs1JSDyKTzsaQi+Q2mrsRvlggk/aBnw1ALwKh8WuVCf4WJDWorygndg9U+J42JNT9hbTu0j49KqCCJxOciOFDfMKvNVitypzNTea5Ldqb/g25O2GoKem+NC3Jdngwrj7vZPz6cOzL5MDh/muKjgL4SdRZ7t3bHiTWPvuGBrxIHzUdokHD5+oCpwJ+Wgyv6efShGQkNdF/BNbREaOtwLLjQ2GAF4w01SFW4OH2a5AL+UywLOkSwE91UCorTjAkNgMnMDUxNVhieoTnRcTfAAnxLPa/V0IAN8qS1CtWt94YJD8qXGj8UrJTl8mOVCuEdgl+LVrqQPijNvU+K4YGvEZB6nSijvXhXE5we2FzTi8nxDDa9OslWGvRkuqiq0W+h5edUVFaCNJpcfFcuO0t8cj2CeC8/3bQOvgbAcPsxzAVjxcCNlfl8rTF2zOy4whEbRi6PKQyZ0k7utFJ0kVUdqohfKx0L5tbNp1UrWN2dm26BmVal+dp4L9dGd5RQmOY+ALsRHV7lQvwZgTPSPrRFzFw27GUWdOC7YGmERuMFWUeVUgpQVKY/ruVffnbz2X//SR1zga2KWC40DC8tnhaURKM5Ck/KEOXyY4kI4wcQPZGyDZpmTjgsMgQlSI4vKpENpMABHsNIQFORU6XFIdDjE6uN+N3XVJBcaA1bC1OcxGEdqAcjhwxQXfh4yF8NWKB7jbZ9I3Y46LtZUYCTdid/ycQwA5ZsVZzWKl5NeeTzjFsmC8OnIMKOj5rjQOLCIcTjmXBDflTPA3J6ktp45fJjjQnhr23bhZzFg/nKFWs5fpYM4LjAEJkhMYCiA9XD3yoQvVAkV4SEoiwt6VrzcJlFmWaJ2GXjABUvHHTJw/KaRGngwVG+/lFUTc1wIAVxgDOG3nhgUnKQkhw9zXGjsuSnblcbzGW5q31xFpEy48N9nP+ob7cEyvxlXe/vSk/duk2/PpY+57VBW2ksu/Fhn87FKXHz0lsFqWR6yImt5zyhuKuWq3rpN3GWONwjt9pkLMM7Cx+pwEVZKWf9Q2ov/ToswsfoNQzo37CMXYGd8Z+VjVbgIHoGVgh+w4r0VedYFPPP8MNdHLhjLefhYCS7K/1CDJR0yOFhw2Gj6QIliJkQYODxyk5JzbVrD1CcuALcIH6vAxT942CZiWYJElQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle \\left(x + 1\\right) \\left(x + 2\\right) \\left(x + 3\\right)$"
      ],
      "text/plain": [
       "(x + 1)⋅(x + 2)⋅(x + 3)"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# simplify 展开乘积\n",
    "simplify((x+1)*(x+2)*(x+3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAOCAYAAAASVl2WAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAXUlEQVQYGWP8//8/Aww0NDQYAdmrgdgYyP4AEmcBMgSA9GwgfgfEJkCsBMRwAFIAUhkKEgGyy4AUyBQ4YIKzcDBGFUAChmA4MIIiCxjEoAgCxQkoLkD0PSA+B8SnAeDgGGsGGU5UAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle 1$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# simplify 使用三角恒等式\n",
    "simplify(sin(a)**2 + cos(a)**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{1}{\\tan{\\left(x \\right)}}$"
      ],
      "text/plain": [
       "  1   \n",
       "──────\n",
       "tan(x)"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(cos(x)/sin(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 4.3 apart 和 together"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "为了处理分数的符号表达式，我们可以用`apart`和`together`函数："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [],
   "source": [
    "f1 = 1/((a+1)*(a+2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{1}{\\left(a + 1\\right) \\left(a + 2\\right)}$"
      ],
      "text/plain": [
       "       1       \n",
       "───────────────\n",
       "(a + 1)⋅(a + 2)"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{1}{a + 2} + \\frac{1}{a + 1}$"
      ],
      "text/plain": [
       "    1       1  \n",
       "- ───── + ─────\n",
       "  a + 2   a + 1"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "apart(f1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [],
   "source": [
    "f2 = 1/(a+2) + 1/(a+3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHwAAAAsBAMAAABVvsF6AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nnOfh0q8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{1}{a + 3} + \\frac{1}{a + 2}$$"
      ],
      "text/plain": [
       "  1       1  \n",
       "───── + ─────\n",
       "a + 3   a + 2"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIIAAAAvBAMAAADdrw/+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n",
      "text/latex": [
       "$$\\frac{2 a + 5}{\\left(a + 2\\right) \\left(a + 3\\right)}$$"
      ],
      "text/plain": [
       "    2⋅a + 5    \n",
       "───────────────\n",
       "(a + 2)⋅(a + 3)"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "together(f2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "简化通常结合分数但不分解："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'f2' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-40-5394828ce0d4>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0msimplify\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m: name 'f2' is not defined"
     ]
    }
   ],
   "source": [
    "simplify(f2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5 微积分"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "除了代数操作之外，CAS的另一个主要用途是做微积分，比如代数表达式的导数和积分。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 5.1 微分"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "微分通常是简单的，用`diff`函数。第一个参数是求导的表达式，第二个参数是求导的符号:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(x + \\pi\\right)^{2}$"
      ],
      "text/plain": [
       "       2\n",
       "(x + π) "
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 4 \\left(x + \\pi\\right)^{3}$"
      ],
      "text/plain": [
       "         3\n",
       "4⋅(x + π) "
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(y**2, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "对于高阶导数我们可以这样："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 12 \\left(x + \\pi\\right)^{2}$"
      ],
      "text/plain": [
       "          2\n",
       "12⋅(x + π) "
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(y**2, x, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 12 \\left(x + \\pi\\right)^{2}$"
      ],
      "text/plain": [
       "          2\n",
       "12⋅(x + π) "
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(y**2, x, 2) # 和上面的一样"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "为了计算多变量表达的导数，我们可以这样做："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [],
   "source": [
    "x, y, z = symbols(\"x,y,z\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = sin(x*y) + cos(y*z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\frac{d^3f}{dxdy^2}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - x \\left(x y \\cos{\\left(x y \\right)} + 2 \\sin{\\left(x y \\right)}\\right)$"
      ],
      "text/plain": [
       "-x⋅(x⋅y⋅cos(x⋅y) + 2⋅sin(x⋅y))"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(f, x, 1, y, 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 6. 积分"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "积分也是以同样的方式进行的:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\sin{\\left(x y \\right)} + \\cos{\\left(y z \\right)}$"
      ],
      "text/plain": [
       "sin(x⋅y) + cos(y⋅z)"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x \\cos{\\left(y z \\right)} + \\begin{cases} - \\frac{\\cos{\\left(x y \\right)}}{y} & \\text{for}\\: y \\neq 0 \\\\0 & \\text{otherwise} \\end{cases}$"
      ],
      "text/plain": [
       "             ⎛⎧-cos(x⋅y)            ⎞\n",
       "             ⎜⎪──────────  for y ≠ 0⎟\n",
       "x⋅cos(y⋅z) + ⎜⎨    y                ⎟\n",
       "             ⎜⎪                     ⎟\n",
       "             ⎝⎩    0       otherwise⎠"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(f, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "通过提供积分变量的极限，我们可以计算定积分:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 2 \\cos{\\left(y z \\right)}$"
      ],
      "text/plain": [
       "2⋅cos(y⋅z)"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(f, (x, -1, 1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "还有反常积分"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( e^{- x^{2}}, \\  \\left( x, \\  -\\infty, \\  \\infty\\right)\\right)$"
      ],
      "text/plain": [
       "⎛   2            ⎞\n",
       "⎜ -x             ⎟\n",
       "⎝ℯ   , (x, -∞, ∞)⎠"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(exp(-x**2), (x, -oo, oo))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "记住，`oo`在SymPy中代表无穷。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 6.1 加数和乘积"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们可以计算加数和乘积使用函数：'Sum'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = Symbol(\"n\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\sum_{n=1}^{10} \\frac{1}{n^{2}}$"
      ],
      "text/plain": [
       "  10    \n",
       " ____   \n",
       " ╲      \n",
       "  ╲   1 \n",
       "   ╲  ──\n",
       "   ╱   2\n",
       "  ╱   n \n",
       " ╱      \n",
       " ‾‾‾‾   \n",
       "n = 1   "
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Sum(1/n**2, (n, 1, 10))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ8AAAAPBAMAAAAIUwCQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nt0ciqgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$1.54976773116654$$"
      ],
      "text/plain": [
       "1.54976773116654"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Sum(1/n**2, (n,1, 10)).evalf()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ8AAAAPBAMAAAAIUwCQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$1.64493406684823$$"
      ],
      "text/plain": [
       "1.64493406684823"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Sum(1/n**2, (n, 1, oo)).evalf()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "乘积也是以类似的方式进行的:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAC8AAAA9BAMAAADPFy0PAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\prod_{n=1}^{10} n$$"
      ],
      "text/plain": [
       "  10   \n",
       "┬───┬  \n",
       "│   │ n\n",
       "│   │  \n",
       "n = 1  "
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Product(n, (n, 1, 10)) # 10!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 7. 极限"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "极限可以用`limit`函数来计算。例如："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPBAMAAAArJJMAAAAAJFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADHJj5lAAAAC3RSTlMAzRAiu5mrdu/dZmiL4QAAAAAjSURBVAgdY2BgEGJgYDDZxMCgEgYkGNhJJVgzdmYB9TEwAACPpQrvlUCHcAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$1$$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(sin(x)/x, x, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们可以用`limit`去检查`diff`函数的推导结果："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKAAAAAUBAMAAAD4uit9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nXd6gLq79C/gHf4qzfkFAyyAAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\sin{\\left (x y \\right )} + \\cos{\\left (y z \\right )}$$"
      ],
      "text/plain": [
       "sin(x⋅y) + cos(y⋅z)"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFMAAAAUBAMAAAADwRznAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$y \\cos{\\left (x y \\right )}$$"
      ],
      "text/plain": [
       "y⋅cos(x⋅y)"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(f, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\displaystyle \\frac{\\mathrm{d}f(x,y)}{\\mathrm{d}x} = \\frac{f(x+h,y)-f(x,y)}{h}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [],
   "source": [
    "h = Symbol(\"h\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFMAAAAUBAMAAAADwRznAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$y \\cos{\\left (x y \\right )}$$"
      ],
      "text/plain": [
       "y⋅cos(x⋅y)"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit((f.subs(x, x+h) - f)/h, h, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "OK!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们可以改变方向，从我们接近极限点使用`dir`关键字参数:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAALBAMAAABv+6sJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKvvZom7mXYyzVQi\n3UQ6SGZXAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAaklEQVQIHWNgYBBgAAIQwaj82YGBIayogYGB\nbQLHLwapDQxTGRg8GRj2J6xkYGA5wACUYP0LJBgcQEyGfBDRAGYm/wNqd2BwZGDgiDE+wMBxgIGd\ngSGcYb4dgytQolxtAwNjvXEAUDncNgBJUBUwaYAbUgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\infty$$"
      ],
      "text/plain": [
       "∞"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(1/x, x, 0, dir=\"+\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACMAAAALBAMAAAAHCCkxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMqvvZom7mXZU\nIkRJD0iWAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAfklEQVQIHWNggAMBEAtMQIUYw74VMDB0Lt0A\nV8LA6cD9iUHoAIMHQqiEgeH8BBUGBvYLDELGIKDCANTA8RmkC6gdCkC8+SAChCEAxJr2j4GBsQAm\nwlDIwMDdm3aBgfsCXIiLgaGLwT+PoQIuwsC4KvIAA+P6tAaEENThAgwMAMSLGqu/gFQwAAAAAElF\nTkSuQmCC\n",
      "text/latex": [
       "$$-\\infty$$"
      ],
      "text/plain": [
       "-∞"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(1/x, x, 0, dir=\"-\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 8. 序列"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "系列扩展也是CAS最有用的特性之一。在SymPy中，我们可以用`series` 函数对表达式进行级数展开:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + \\frac{x^{4}}{24} + \\frac{x^{5}}{120} + O\\left(x^{6}\\right)$"
      ],
      "text/plain": [
       "         2    3    4     5        \n",
       "        x    x    x     x     ⎛ 6⎞\n",
       "1 + x + ── + ── + ── + ─── + O⎝x ⎠\n",
       "        2    6    24   120        "
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "series(exp(x), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "默认情况下，它在$x=0$附近扩展表达式，但是我们可以通过在函数调用中显式包含一个值来扩展$x$的任何值:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle e + e \\left(x - 1\\right) + \\frac{e \\left(x - 1\\right)^{2}}{2} + \\frac{e \\left(x - 1\\right)^{3}}{6} + \\frac{e \\left(x - 1\\right)^{4}}{24} + \\frac{e \\left(x - 1\\right)^{5}}{120} + O\\left(\\left(x - 1\\right)^{6}; x\\rightarrow 1\\right)$"
      ],
      "text/plain": [
       "                         2            3            4            5             \n",
       "                ℯ⋅(x - 1)    ℯ⋅(x - 1)    ℯ⋅(x - 1)    ℯ⋅(x - 1)     ⎛       6\n",
       "ℯ + ℯ⋅(x - 1) + ────────── + ────────── + ────────── + ────────── + O⎝(x - 1) \n",
       "                    2            6            24          120                 \n",
       "\n",
       "        \n",
       "       ⎞\n",
       "; x → 1⎠\n",
       "        "
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "series(exp(x), x, 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "可以明确定义级数展开的顺序为:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle e + e \\left(x - 1\\right) + \\frac{e \\left(x - 1\\right)^{2}}{2} + \\frac{e \\left(x - 1\\right)^{3}}{6} + \\frac{e \\left(x - 1\\right)^{4}}{24} + \\frac{e \\left(x - 1\\right)^{5}}{120} + \\frac{e \\left(x - 1\\right)^{6}}{720} + \\frac{e \\left(x - 1\\right)^{7}}{5040} + \\frac{e \\left(x - 1\\right)^{8}}{40320} + \\frac{e \\left(x - 1\\right)^{9}}{362880} + O\\left(\\left(x - 1\\right)^{10}; x\\rightarrow 1\\right)$"
      ],
      "text/plain": [
       "                         2            3            4            5            6\n",
       "                ℯ⋅(x - 1)    ℯ⋅(x - 1)    ℯ⋅(x - 1)    ℯ⋅(x - 1)    ℯ⋅(x - 1) \n",
       "ℯ + ℯ⋅(x - 1) + ────────── + ────────── + ────────── + ────────── + ──────────\n",
       "                    2            6            24          120          720    \n",
       "\n",
       "            7            8            9                      \n",
       "   ℯ⋅(x - 1)    ℯ⋅(x - 1)    ℯ⋅(x - 1)     ⎛       10       ⎞\n",
       " + ────────── + ────────── + ────────── + O⎝(x - 1)  ; x → 1⎠\n",
       "      5040        40320        362880                        "
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "series(exp(x), x, 1, 10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "级数展开式包含了近似的阶数，当我们对不同阶数的级数展开式进行计算时，这对于跟踪有效性的阶数非常有用:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMQAAAAwBAMAAAC8i8hXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRK5CYII=\n",
      "text/latex": [
       "$$1 - \\frac{x^{2}}{2} + \\frac{x^{4}}{24} + \\mathcal{O}\\left(x^{5}\\right)$$"
      ],
      "text/plain": [
       "     2    4        \n",
       "    x    x     ⎛ 5⎞\n",
       "1 - ── + ── + O⎝x ⎠\n",
       "    2    24        "
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s1 = cos(x).series(x, 0, 5)\n",
    "s1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAF8AAAAcBAMAAAD1rn4EAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n",
      "text/latex": [
       "$$x + \\mathcal{O}\\left(x^{2}\\right)$$"
      ],
      "text/plain": [
       "     ⎛ 2⎞\n",
       "x + O⎝x ⎠"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s2 = sin(x).series(x, 0, 2)\n",
    "s2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAF8AAAAcBAMAAAD1rn4EAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACCUlEQVQ4EY1Sv2tTURT+Xsx7SV5+9FKh0MnX\nCm61obhYB99SHBv8A0wUwbFO6qJDUcShtCiCwcEIuuhglg7i0Afi4CBJ7SDSwS4OTiYiCCLEc867\n971cSfFdOOd833fOx7l5N0DG4+3vZpzUY5dxOpvhnR67jV6Yy2CpdPXQHaw23Pr/HSWVzLxSuJKQ\nf4C7O7P5ULSPaWcdKKb2VGe04G5/GDUI5IdJw+sDOYpJ50hwQWFxi1o8pc9NqvmBYVL9QNM95ysw\nxc1aR0twW24E3DVUqjG4w2oXKPFtespMvH/xkvC+oZah2PeJl3nDrWTg+2hE+CTF9PGVE4HoZkMz\nYlo+oPSZwpk/df4SVT5NBadVWC9uCTOGL3KPWp/ENxQzuBHckwG6YgRPeUM/FG4MO8J6DSoPKC5i\nTT0TCfTacFDqMvPa7Uff2u06wSeKhdecNigUHjOUs9qhMhUIBsyGsxEJlZ+s8gbgh2ROtEHuFQvG\nsMZrZrkVGyp/4j7lXghfLSInd0g2HPtFb7otQ2coP80NMKct9JWanR2ci6nZUBgtr+yFotF3d3+X\nB/mr2kDvcnR++pOmxoC3mxtRPMHffWlu4bqeP+SlTZdqTX5JKtxPISEntCiTasuSnPTfbukpqRyk\nmFChbtFJZNkSi5FFJ5FZS7xmsYnE74/J+e4YOQw+H2tU1RjJCv8Cl4Bl3Hr5MZEAAAAASUVORK5C\nYII=\n",
      "text/latex": [
       "$$x + \\mathcal{O}\\left(x^{2}\\right)$$"
      ],
      "text/plain": [
       "     ⎛ 2⎞\n",
       "x + O⎝x ⎠"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(s1 * s2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "如果我们想摆脱阶数信息，我们可以使用`removeO`方法:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAG8AAAAwBAMAAADtMzlxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRK5CYII=\n",
      "text/latex": [
       "$$\\frac{x^{5}}{24} - \\frac{x^{3}}{2} + x$$"
      ],
      "text/plain": [
       " 5    3    \n",
       "x    x     \n",
       "── - ── + x\n",
       "24   2     "
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(s1.removeO() * s2.removeO())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "但请注意，这不是$\\cos(x)\\sin(x)$到$5$的正确扩展:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANsAAAAwBAMAAABqLhIyAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$x - \\frac{2 x^{3}}{3} + \\frac{2 x^{5}}{15} + \\mathcal{O}\\left(x^{6}\\right)$$"
      ],
      "text/plain": [
       "       3      5        \n",
       "    2⋅x    2⋅x     ⎛ 6⎞\n",
       "x - ──── + ──── + O⎝x ⎠\n",
       "     3      15         "
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(cos(x)*sin(x)).series(x, 0, 6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 9. 线性代数"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 9.1 矩阵"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "使用`Matrix`类可以定义矩阵："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [],
   "source": [
    "m11, m12, m21, m22 = symbols(\"m11, m12, m21, m22\")\n",
    "b1, b2 = symbols(\"b1, b2\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGgAAAAyBAMAAABCJ4MDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhBEqyJ2u93NZplQnf8bAAABo0lEQVRIDe2WsUrDUBSG/zZprNaKqIuT2cSlUnyAKta9g3uL+ADiplPBxc1OgpuLgosUn6Dgg+jm0A5VsHaL99xr7vUkp0PqJPTAhXD/+5Hwk3wEG9EHMs129I7V+kEmBof1KtayIXR6bwaZ0nQRweblzVmwdS/1KGUayi88oHmNKuDfmuVwnlUeGyrS0H6ph6sQL/Dfuno5BizLt+d6MdRotlBr4BzwumY5imWltj+OIdSAPrAuQSzLtYojCw2AJwTjgnAnsAzep4VWUBhhqXchQSxDqRND6p7lIRbDUIB4hopiTHtBF/Mn8E53BYhn5VcL0YWZdHtxYko6wrHaSHwa3rIK1ZJG7fvPO3cpqNj/6tCSGNrPRdEwBUmHhb3E4wknhK0Z9FPKrIg/FyEJMX7lpExX7oRIMiRhuuGZJEuSIQnz11hZUibKUsuQPnk3VpaUibLUMuSQlSVlsixJhgnIylKLUpIlyTABWVlqUUqyJBlyyMlSi1KSJcmQQ06WlImyJBlyyPVImSRLLcMJsqRMlCXJcJIsKfsvspzq122an8Rv50XFB+Ww4SIAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}m_{11} & m_{12}\\\\m_{21} & m_{22}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡m₁₁  m₁₂⎤\n",
       "⎢        ⎥\n",
       "⎣m₂₁  m₂₂⎦"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = Matrix([[m11, m12],[m21, m22]])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}b_{1}\\\\b_{2}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡b₁⎤\n",
       "⎢  ⎥\n",
       "⎣b₂⎦"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = Matrix([[b1], [b2]])\n",
    "b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "有`Matrix`类实例的基础上我们可以做常规的矩阵代数运算："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}m_{11}^{2} + m_{12} m_{21} & m_{11} m_{12} + m_{12} m_{22}\\\\m_{11} m_{21} + m_{21} m_{22} & m_{12} m_{21} + m_{22}^{2}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡    2                               ⎤\n",
       "⎢ m₁₁  + m₁₂⋅m₂₁    m₁₁⋅m₁₂ + m₁₂⋅m₂₂⎥\n",
       "⎢                                    ⎥\n",
       "⎢                                 2  ⎥\n",
       "⎣m₁₁⋅m₂₁ + m₂₁⋅m₂₂   m₁₂⋅m₂₁ + m₂₂   ⎦"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJEAAAAyBAMAAACufiRQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nDbOYk3PHxJxH96vsyH4p/gHeHTH4ApAfnQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}b_{1} m_{11} + b_{2} m_{12}\\\\b_{1} m_{21} + b_{2} m_{22}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡b₁⋅m₁₁ + b₂⋅m₁₂⎤\n",
       "⎢               ⎥\n",
       "⎣b₁⋅m₂₁ + b₂⋅m₂₂⎦"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A * b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "计算行列式和逆，诸如此类:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKYAAAAMBAMAAAAaIdvMAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n9AyNtKqM6qgiUQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$m_{11} m_{22} - m_{12} m_{21}$$"
      ],
      "text/plain": [
       "m₁₁⋅m₂₂ - m₁₂⋅m₂₁"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.det()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{m_{22}}{m_{11} m_{22} - m_{12} m_{21}} & - \\frac{m_{12}}{m_{11} m_{22} - m_{12} m_{21}}\\\\- \\frac{m_{21}}{m_{11} m_{22} - m_{12} m_{21}} & \\frac{m_{11}}{m_{11} m_{22} - m_{12} m_{21}}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡       m₂₂               -m₁₂       ⎤\n",
       "⎢─────────────────  ─────────────────⎥\n",
       "⎢m₁₁⋅m₂₂ - m₁₂⋅m₂₁  m₁₁⋅m₂₂ - m₁₂⋅m₂₁⎥\n",
       "⎢                                    ⎥\n",
       "⎢      -m₂₁                m₁₁       ⎥\n",
       "⎢─────────────────  ─────────────────⎥\n",
       "⎣m₁₁⋅m₂₂ - m₁₂⋅m₂₁  m₁₁⋅m₂₂ - m₁₂⋅m₂₁⎦"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.inv()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 10. 求解方程"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "对于求解方程和方程组我们可以用`solve`函数："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ -1, \\  1\\right]$"
      ],
      "text/plain": [
       "[-1, 1]"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(x**2 - 1, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ - i \\sqrt{- \\frac{1}{2} + \\frac{\\sqrt{5}}{2}}, \\  i \\sqrt{- \\frac{1}{2} + \\frac{\\sqrt{5}}{2}}, \\  - \\sqrt{\\frac{1}{2} + \\frac{\\sqrt{5}}{2}}, \\  \\sqrt{\\frac{1}{2} + \\frac{\\sqrt{5}}{2}}\\right]$"
      ],
      "text/plain": [
       "⎡       __________        __________       ________      ________⎤\n",
       "⎢      ╱   1   √5        ╱   1   √5       ╱ 1   √5      ╱ 1   √5 ⎥\n",
       "⎢-ⅈ⋅  ╱  - ─ + ── , ⅈ⋅  ╱  - ─ + ── , -  ╱  ─ + ── ,   ╱  ─ + ── ⎥\n",
       "⎣   ╲╱     2   2      ╲╱     2   2     ╲╱   2   2    ╲╱   2   2  ⎦"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(x**4 - x**2 - 1, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "方程组："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAH4AAAAVBAMAAAByPkciAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n/1sDfgDNr2XH9LJoSQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left \\{ x : 1, \\quad y : 0\\right \\}$$"
      ],
      "text/plain": [
       "{x: 1, y: 0}"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve([x + y - 1, x - y - 1], [x,y])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "在其他符号表达方面:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAOAAAAAmBAMAAAAvsop7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n4r9Q1gIGAlZHdQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left \\{ x : \\frac{a}{2} + \\frac{c}{2}, \\quad y : \\frac{a}{2} - \\frac{c}{2}\\right \\}$$"
      ],
      "text/plain": [
       "⎧   a   c     a   c⎫\n",
       "⎨x: ─ + ─, y: ─ - ─⎬\n",
       "⎩   2   2     2   2⎭"
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve([x + y - a, x - y - c], [x,y])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 进一步的阅读"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* http://sympy.org/en/index.html - The SymPy projects web page.\n",
    "* https://github.com/sympy/sympy - The source code of SymPy.\n",
    "* http://live.sympy.org - Online version of SymPy for testing and demonstrations."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 版本"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/json": {
       "Software versions": [
        {
         "module": "Python",
         "version": "2.7.10 64bit [GCC 4.2.1 (Apple Inc. build 5577)]"
        },
        {
         "module": "IPython",
         "version": "3.2.1"
        },
        {
         "module": "OS",
         "version": "Darwin 14.1.0 x86_64 i386 64bit"
        },
        {
         "module": "numpy",
         "version": "1.9.2"
        },
        {
         "module": "matplotlib",
         "version": "1.4.3"
        },
        {
         "module": "sympy",
         "version": "0.7.6"
        }
       ]
      },
      "text/html": [
       "<table><tr><th>Software</th><th>Version</th></tr><tr><td>Python</td><td>2.7.10 64bit [GCC 4.2.1 (Apple Inc. build 5577)]</td></tr><tr><td>IPython</td><td>3.2.1</td></tr><tr><td>OS</td><td>Darwin 14.1.0 x86_64 i386 64bit</td></tr><tr><td>numpy</td><td>1.9.2</td></tr><tr><td>matplotlib</td><td>1.4.3</td></tr><tr><td>sympy</td><td>0.7.6</td></tr><tr><td colspan='2'>Sat Aug 15 11:37:37 2015 JST</td></tr></table>"
      ],
      "text/latex": [
       "\\begin{tabular}{|l|l|}\\hline\n",
       "{\\bf Software} & {\\bf Version} \\\\ \\hline\\hline\n",
       "Python & 2.7.10 64bit [GCC 4.2.1 (Apple Inc. build 5577)] \\\\ \\hline\n",
       "IPython & 3.2.1 \\\\ \\hline\n",
       "OS & Darwin 14.1.0 x86\\_64 i386 64bit \\\\ \\hline\n",
       "numpy & 1.9.2 \\\\ \\hline\n",
       "matplotlib & 1.4.3 \\\\ \\hline\n",
       "sympy & 0.7.6 \\\\ \\hline\n",
       "\\hline \\multicolumn{2}{|l|}{Sat Aug 15 11:37:37 2015 JST} \\\\ \\hline\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "Software versions\n",
       "Python 2.7.10 64bit [GCC 4.2.1 (Apple Inc. build 5577)]\n",
       "IPython 3.2.1\n",
       "OS Darwin 14.1.0 x86_64 i386 64bit\n",
       "numpy 1.9.2\n",
       "matplotlib 1.4.3\n",
       "sympy 0.7.6\n",
       "Sat Aug 15 11:37:37 2015 JST"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%reload_ext version_information\n",
    "\n",
    "%version_information numpy, matplotlib, sympy"
   ]
  }
 ],
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