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": [
    "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": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sp"
   ]
  },
  {
   "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": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "sp.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": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = sp.Symbol('x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "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": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(pi + x)**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 定义符号的可替代的方式\n",
    "a, b, c = sp.symbols(\"a, b, c\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sympy.core.symbol.Symbol"
      ]
     },
     "execution_count": 17,
     "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": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = sp.Symbol('x', real=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x.is_imaginary"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = sp.Symbol('x', positive=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\text{True}$"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 21,
     "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": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 1 + i$"
      ],
      "text/plain": [
       "1 + ⅈ"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1+1*sp.I"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABgAAAAOCAYAAAA1+Nx+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAiElEQVQ4EWP8//8/A7VAQ0ODEdCs1UBsDGR/AJnLAiIoAUCDBID6ZwPxOyA2AWIlIIYDalgAcmkoyESgZWVACuQLOGCCs2jEGLWAYMDSPIhYoMlsL9ApoORGLAgF6jtHjGKQBaBkZkyMYnLU0DyIhr4FjNQo7IDxCCrgQIkEVBaB6HtADEoEpwEsISASVdEGngAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle -1$"
      ],
      "text/plain": [
       "-1"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.I**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "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": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(x * sp.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": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "r1 = sp.Rational(4,5)\n",
    "r2 = sp.Rational(5,4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{4}{5}$"
      ],
      "text/plain": [
       "4/5"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABUAAAArCAYAAACXZ8NLAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACg0lEQVRIDe2X320UMRCH91AKiKCDSwcJqYDQQSIqgHRAxNPdGwodEEqADpIOIq4DQgWQ6+D4PmvH8nr3dpUcPJGRrLHnz88z9njtnW02m2aIlsvlIfJz+PmI/iu6I2zWpc1eOaj6OqxKGc77jL/QftNe0ua0Hg2C4vy+Z4mgjehMXWtjNj16Vksw1tB0OinVdmPjHijGbwC+GnOa0nVA25Q+TzlN6TMogHOM1/C7KacpfQbF0PLZKe2YLIEC9g7Bzmln0Dbt/b+RdoBap67lMaAWe0mW1ryV38EvSuVYfw/jGwxsHUJ+j+AGnoq9o5wYlBtVm3okbQ+mWf1BITI3zHMdR9AsrpF/Eh3uMjmZNnJL0G/Ebdj0QFHuTGPpPxr8CfTRS7fV8T9f09lisRi+o7eu2LTi6URNr9FDLf5JneZnD9/CORHFleG30vfSBfLOe8qokV3KoV+0A9olsny1p91HIKCKfHW0jr6pXtPP1w3978g+wr/BG7gfamXaJeBI35nf0jJhYNRrWr4QkXmVe/MmQI3pa+M4X/EBeoLwHgNnLckIBTETyUx6y4HslnYS/gGqs9ewsw5RTObkrnVN4ae+SRsFWF7LyjpdfuhXEUWlr4fPFUSktbIBJD0mUERFJAfGEVXpE9GnjLaC4uEGXQGeruYSYaT/Qt0gKEDupK+T8iciohnCjCys2z4oQJZNUwE6jrRTitoUFLJOnSY9jqd0DkpA+j7S5i2AVRL9VpRYRJoOSU4fRzfG119sTDg5UaTuOnuEazpCYIWkbMpjeo0izVR4GMEhxp7vRPR/0PGbUB7Tn8heIVtplOoUbgSmldYTXlIyLARG9QGAWIZjxhlQuz/LF9u42ZG+RwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle \\frac{41}{20}$"
      ],
      "text/plain": [
       "41\n",
       "──\n",
       "20"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r1+r2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABUAAAArCAYAAACXZ8NLAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACtUlEQVRIDe2X31EUQRCH7ywDQDPwyAAwAjUD1AiQDKR8unuzIIODCAAzOIzAggwgA/UyOL9vbnpqdr2FXco3mare6enp/k1Pz7/e8Wq1GtVlNpvt0L6EduGXdV/N03dct+HPkd0oe+6HxhbVGfQL2oMm0MaCrn0L6BD+SiVqnZC2bQeoHr1XgMJnKr3tKhpfoZcAs5JO3YVBAo3GQzVA++g44JtaF/m7uv2sbvTgv6CzBKQz1mIM8hR9vbwB1Poj9BMyjpfISjh6g2Jk3KLs0T6KBvxv6AD6pmzo9LXZwfhUpioX8GfI08C9QTGIOJZVrkCv4QV0Ow72VMAA175d3MOPAk1TbKPldppF7+lnI49m8qYFukvbrZZ2wCDQbORpKuceXs8/QAcx0DguFDo9fhFsa6fiBfGDvhPqUjJohOElHV+RpctEpQJaLP4BM2j6fcd7Au0bqf56/3lMx9PptPlG9w9dp+bTieoMzaM7Bu9Trjif50ZBtgVNQlie6CyMZ9cHzLzqCHm5J7PRd0HgQx73qrd/KsnTDHhMbdIlqaDRNfzbtWr5OpgXuB4L6FvfyBDDU5+H8hzAjwDTy0+wvggvlOVihpKSuRC064ip3phlxFRCz4esEa/ouK8OUI3vAF12KLcH61Bbi+89UQxyi9qEehwo8IbDpDcGMkGbI4+F604mUHIh3CaxI2BTEeyC/hMJ3vVoLqhP9CbiormF5pv62jL0FuqHPGKaXIgPHszhTRoOQ/ZA7RYzTM7s7+nT4TYabQJEtoDM8LpKinPDUwzM6bdrQPjiAX2etFigGtgsZYRuWqwCisCFeU3dXhgH8hRZTulPvzXrZvm6z0t6nrYUisbCbVI6sroemDknoKznSSuxhvcXyR+MV/BL7QLUOOnppuKxLJcFvA7EbBzUWThQAhTgD4e8L3Lsd3RVAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle \\frac{16}{25}$"
      ],
      "text/plain": [
       "16\n",
       "──\n",
       "25"
      ]
     },
     "execution_count": 30,
     "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": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 3.1415926535897932384626433832795028841971693993751$"
      ],
      "text/plain": [
       "3.1415926535897932384626433832795028841971693993751"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.pi.evalf(n=50)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "y = (x + sp.pi)**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "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": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.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": 34,
   "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": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y.subs(x, 1.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 21.5443823618587$"
      ],
      "text/plain": [
       "21.5443823618587"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.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": 37,
   "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": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y.subs(x, a+sp.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": 38,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [],
   "source": [
    "x_vec = np.arange(0, 10, 0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [],
   "source": [
    "y_vec = np.array([N(((x + pi)**2).subs(x, xx)) for xx in x_vec])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "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": [
    "import matplotlib.pyplot as plt\n",
    "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": 43,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = sp.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": 44,
   "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": 47,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "11.4 ms ± 82.5 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "\n",
    "y_vec = np.array([sp.N(((x + pi)**2).subs(x, xx)) for xx in x_vec])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.41 µs ± 17.3 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)\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": 49,
   "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": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(x+1)*(x+2)*(x+3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\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": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.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": 51,
   "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": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.sin(a+b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "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": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.expand(sp.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": 53,
   "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": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.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": 54,
   "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": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# simplify 展开乘积\n",
    "sp.simplify((x+1)*(x+2)*(x+3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAOCAYAAAASVl2WAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAXUlEQVQYGWP8//8/Aww0NDQYAdmrgdgYyP4AEmcBMgSA9GwgfgfEJkCsBMRwAFIAUhkKEgGyy4AUyBQ4YIKzcDBGFUAChmA4MIIiCxjEoAgCxQkoLkD0PSA+B8SnAeDgGGsGGU5UAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle 1$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# simplify 使用三角恒等式\n",
    "sp.simplify(sin(a)**2 + cos(a)**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "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": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.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": 57,
   "metadata": {},
   "outputs": [],
   "source": [
    "f1 = 1/((a+1)*(a+2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "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": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "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": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.apart(f1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [],
   "source": [
    "f2 = 1/(a+2) + 1/(a+3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{1}{a + 3} + \\frac{1}{a + 2}$"
      ],
      "text/plain": [
       "  1       1  \n",
       "───── + ─────\n",
       "a + 3   a + 2"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{2 a + 5}{\\left(a + 2\\right) \\left(a + 3\\right)}$"
      ],
      "text/plain": [
       "    2⋅a + 5    \n",
       "───────────────\n",
       "(a + 2)⋅(a + 3)"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.together(f2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "简化通常结合分数但不分解："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{2 a + 5}{\\left(a + 2\\right) \\left(a + 3\\right)}$"
      ],
      "text/plain": [
       "    2⋅a + 5    \n",
       "───────────────\n",
       "(a + 2)⋅(a + 3)"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.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": 64,
   "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": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "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": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.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": 66,
   "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": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.diff(y**2, x, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGEAAAAaCAYAAACn4zKhAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAEYklEQVRoBe2Y7VEbMRCGjYcCIOkAOoC4gpgOAqkA6CAe/7L/ZUgHJBUwSQeQCoB0EEog7oC8j5BudDqdThf7DpzxzsiSVivtl3ZX562np6fBBrq3wHw+3xGXqeW0Z/tT4Rfb3bPfcLAWuJDBz501NL7U+F5tf+iQm75zC5zJ8GOPy4XGe8IdbJzgWaXjIVFwF+OxtakJMbN0j1MEEAkf1L9MOhJjV5i617YnDm10Eu0BDlA7RLze05EE+CS+CPG/Afkd3ZJgnUUUHGq8gLiUjoTEON99Aogc2AMmdv5O/aPaRPhfjibViw7vj9S7M1Lka7cmvXDCg/ofMeGFJwPwSjpmXT32XmzNZjPer9/UMCiGZWFXBMZLGhvQvHQASOHwKIyPNL4BVwdah89P9SYE6+jWHS/9eHa+Vx+zH89S/wJSrCdDiNWO1UBcqdUBBj/1F7WHA2FG9DQB+xGiN5B8Y7Wz3hg+M0JHdA0B5/BEpXeNZ2urjzUO+KNNYZQQAVR5cuKDxnVwovXiY6WOaMV4oo/WG0jHr2rYiTRdRIPGu3VCtCnMGJt8VxwcHFqrrPbgwJSDgqPWfoquJ7labOcSypCmmETozUtH66nizN7amqG9ONCljZHGpD1q0Ec14FY00WL3vNzNr3hS76ZqtRdMa9z8MMLR9Yg1tUZoEwmVw8QcB2Asv9hU6ISg4P+OLVgcL4YvNM1v1XgokM85FwPEcqzQ3YF4w/OtGheIhmw8KkwxtWOemaEDtGR0xS5ZsJQTxIGCzE1AwBRgSF5fFdBeIiA0Mk9Zd/PfRNYr56wSIZm4XEQfeZ1bjUEvNSba0eWGsZ1rWgF0TUVPaUN2Oirt0kQC8ApAmNhNCMkx5CJE2vmdzvDrBWGMggannluYBNEgC3UnBPgOtB6TER7Rs4XH2DQHPMHdRRsJ2ZRmkD07Ev7JCRLI5G/1MeWc4Fm9VdinxZhN6c2nH9TJITwRxavNGbC0L2eivUSFf4EOhPPnsWNSl65C3zodWcX21RcO0BhFU57nZjSGp85wt9mlooFwO7SK5P0h0PPKY5fS05EhLykpC1o5QcbgVsT+duDGpZiyVhHeGvhavTO+cazmfnqaat5087KUbUuEfNpD1Jdedp68dUcSCb4OdXQGn52OxBgjUoipA+RgBzAkRFMhT34ll4aA8WkUQHd7CoMLxxqvpZcCHEDtKGTSnDF1q+QYzX3gFXXtI1Jj4wQxwbgYgackcC8chuOF4IwLDY4w9UC9D9CmAIF8xzlaFKHIYeyBeJ2rERnQ8mn/qHGRmqDpGTD254BnyviOFH3CF59bq/Slf1ErqytEyJh8J/AfVZPDVsZVvJYuzG2FEU8uKhdpP3fvMJdwBXTcjKKYr+C8nCNIHX4qydmzLA0vu+wogFlvkQAz3Q7SEiknu2ixb11AehEFfNSRxrKhz0hAKD6OYrUhW+BXTohu0Q/AlNy9RgKC2NsyVt/01ZmS+9WtSR/zlFXfOsr/AobBiookRUGSAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle 12 \\left(x + \\pi\\right)^{2}$"
      ],
      "text/plain": [
       "          2\n",
       "12⋅(x + π) "
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.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": 68,
   "metadata": {},
   "outputs": [],
   "source": [
    "x, y, z = symbols(\"x,y,z\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = sp.sin(x*y) + sp.cos(y*z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\frac{d^3f}{dxdy^2}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "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": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.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": 71,
   "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": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "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": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.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": 73,
   "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": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.integrate(f, (x, -1, 1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "还有反常积分"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "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": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(sp.exp(-x**2), (x, -sp.oo, sp.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": 78,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = Symbol(\"n\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "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": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.Sum(1/n**2, (n, 1, 10))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 1.54976773116654$"
      ],
      "text/plain": [
       "1.54976773116654"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.Sum(1/n**2, (n,1, 10)).evalf()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ8AAAAOCAYAAAAxIc4GAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAGU0lEQVRoBa2a7XUVOQyGLzkpIJutAOiAjw5CB8tSAdABHP7xjwMdABWw0AF0wEcHsBUQ0kH2eSaW1zPjmdFN0DmObfm1ZMmy7Llw7fz8fBf0/PnzW7TfU27TPgt+tmbOywn2HbxvE17tFn2PqR9XZqexhmPsdTPlmPZDeLO1w7vBWOg5om3/JfxP1IvEuD7prpGxsPcnmJsU5f1YEtbgA9L1TwYHRhueUdT9J8X+e/iXsmcfeWD13VOKdIdySnkKf7TXBbfo88Oi9G0RoCAF70VFyUcmuUmD8dQGscVNWSLHRwteAM5wZd1fwb+m/cp51AbKv9QenhoEtN0YnROO2NF+BO8j9X3KB9pLNNMtkDnqfhFzqdXxlfoepeouWH266R/mpXDKhAz0ao8M+gbfEWVve7LykO0a1X1fnRJtD2HYHvu/6fMDJp4piKIh7xR2CXKDPiGjPXUqH21CKxfsk7a/1F7BeWCOGR8Cz/m0DeQvlDiVsiUD7RHjfw29iz//lLbZo0vgu2uEr7zRJsM7g+emt5k45Gb9k8IV/Qb/lB7CuKw9WXkGmnoqsR79rf2uP2jT5weBvGyNYjfUjDPacPhmgHs9ufDFu1jLIm3g1NsLbgNQw1uSN9KH7Kvo9tT3MvZn+CfIPgrltFP+yeKKXG+TmnlC11qN/DWf7yPvBD2/WhuLXhOPB9LMKG36/MrBhxJPmtlzdTNdTUMPwL9p+kvNLq4x/LQz8ac8MDp7INpm5T+sG15kwV6mEtbVXea7AT3d4QPHg7L+yeKU6xVukPtsqIEOz6x0GXv2kacPf6A3bKU7omE9+pqy6vPD0bTLddzkbyiyfkBx8z1J3ccvOK+yJQcxdEFrOMYMdoHHF+jRXx/fUm9sGGCuweFG+UadHQJ4i2tkrN3sQV7nT6tbv2T8k8XtWIMb6xXvATILeevoc4Nx9t6Dt2gPc3aMp+WBXcq4rl9ZvRtB/sznVwo+BLYbcYd+vXpp6xS/PKszaJuSDZzedenaB0rilKtBUxqcADPSfx1HrmPOuUvRSV8oI0roPi4Teic/smGc/pR/0JnCtQtlju90D5CBZa091iNK2DPgs/JGwkuHufpVf9f9D1wZ6/r8IEBXrG+hZJpBfNC/hd86tptpOrozuOHRi/wagMXQsyIvAqGKZ9wM9Iri6X1H8Qstrt/AZXQHdqmO7BvjWf9kcbtm3WY8r0ID4Dv86o+iPGXPHvKK2FHlh8YbZNSPvxiFt+jzKwUfgmOje5nMrycD744LAetHQOa6zeLUfZ1iBnhC8dR7+nz0S701XYzwF7yZUxnDzxMOwMvongW1cwtFVox3Z8o/6E3hQklZpx90/nzk+8sPu7gO97Vnx3zt3pQX+tuaue6p1/boZ58WE20wI58fxsAVajc5nNcTcwOlBoVfQlsBkcKFEuSpd2Q0vAjwqgueWWFH7dXUkteumeKkjGXWGO/No1ZQaQev6oa/6Z8yN4sT7kHz4FVi/R8oZsHvlLQ9RcCmPHAGzojQZ9DuqEd7UHibPv9dwWfQLJFOdfwui2x/BxLvAg1O+eL86trEgZ+9LZgXZKb1JA4HgtqAMAvvaPv1NfDtTyi1Ruar22uuZ3NkvvpVDS7sn6irXcelFA792uMhmdkBzyxokLiOlD3gXiTlAfuf0ONz5SZ1DTza4RNvh02f/47g89QYNFO6DcMsERsRdcUx9ouOgRJXhmMpHHM0/i3lOu1hI6g13oBW90COlfEakDFGPTwJqGMspRu8h0W7p6Re3zhtYGT9k8KFPdQe2gjcdh0Gp/Y4lrKnyFyV1yoAr49NEtMk4J749kv5/KAVutZGoKftnDJEdGDpa6DG1s2grQP+pox+CY85TS3OskU9nIE2fX955frAnl6vOml0QMDoKOWKP6Neoplu8H5cnRYZwzza4mY2w0/5J4sri/Sw1rdd4e2Q4TXozy29oAzYzB4G0vKQrd89fMaD/7QZRV7ry02fX/M/FiBgEMZkM4GLc/Fu4GfG6hcMbd8TBlpNtfQHgmfwOVcy7b+ANw2CYRC+QaIuT5DkBum0qkvmFo7xGvDA1e2GzE57keXbrs2wOtF/o1zCr66Reep7Rhk+LqjvUtZsTvmn2LTpR3D6Tv3tAVyzf8uelDz0mnxi32iOyKzf3jqrPv8Ph9kUEtUPvTAAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle 1.64493406684823$"
      ],
      "text/plain": [
       "1.64493406684823"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.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": 83,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\prod_{n=1}^{10} n$"
      ],
      "text/plain": [
       "  10   \n",
       "─┬─┬─  \n",
       " │ │  n\n",
       " │ │   \n",
       "n = 1  "
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.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": 84,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAOCAYAAAASVl2WAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAXUlEQVQYGWP8//8/Aww0NDQYAdmrgdgYyP4AEmcBMgSA9GwgfgfEJkCsBMRwAFIAUhkKEgGyy4AUyBQ4YIKzcDBGFUAChmA4MIIiCxjEoAgCxQkoLkD0PSA+B8SnAeDgGGsGGU5UAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle 1$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.limit(sp.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": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKAAAAAVCAYAAADByOXrAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAGXUlEQVRoBeWa0XEUORCG51wbgDlHcEsGgCPAZGBwBEAGpniy3yiTATgCDjKADMDOwGRwsBn4/k9WC60saTXjndm9o6t6NSO1ult/t3o0Y3fX19fdtvDJycl8Cl+msjPFWtZlYwpMcjZ2ui2h09PTY7nyYCJ35t7eROa228yE2N/CvSkB5SATf4pfjAGl9B5K757aT2PoT3XKzhf6vN10+Le69xhMgn0O96YEVETm4l3xw3VHR06h97XaV+vWXdMne2+9Xez/lrQJ7FPcmxJQk6gY99S+HCFSZ9L5bgS9LSqxi/1BJDwOxKM8FQY51H/SprAPuDclIOsS0Iv+62ua8Uy63zdJrlnI28X+0CrIvKFz17yaQeo2gn2Me3MCDlreikly5EAi31eIjT2M/WdjG9k2/VuAvcN9BjByhrdPkoEqZzv6SP0PxXP1fRTTftH9U7WdWubQ/6f4bzHlnJcJaF/8VTKcs2qELvdCkBPSfHyxRxw6n4vx40gMYcO9uESy99V/Fdv2Y+dqne9u5q8f7D8RT16FvV/gBu5GYU10RDJXXoD1fVR/wE3Xxfj5ObmmFfsmPL0P5AP0VPeXYmJ1oDaHrcN9pkGCfKaWIATSvTvvqSVTSUQDwMno/lIX99X+VEsSHuraJZxaALlQ+0lcq3CPJFc7/+GX+cFnmnMxAXoltnOEvTm7Fxn1swmQi5OfCmebQ5dLxLrYfJOS/CQ4F+LnurZNhB8k13cxAQRHgvqEPrWOdP2ZMfFbcTV+NifTrsK+L55gTkz2xMTts1ryo/Te4HDfkRCOPPIL0WUgC6x1BACsw7ff1JLlIeC6JjmhVYEFvB9OMvmRDipf+oJAEplfJL0blyzB/CqG2Eipr/SFioFQRNjHj6mJxCLRbD3YNz8W3hlkcpuYLwYEmQRtjZ9XGZoa9r3w9Pg/VsuGoDiAN592SsmHEw53KiCPVW74zkeQyFwW3eezCEmYIwM0N0YfSWRgpzLf5EOcSCyKquD61MaP0x+6t0BS7d4kytgIaZ+JoA/AiyTd7OzcZsL/TuM5oPE19jHoVz+4kDxh0zKoftbg1qFrxuNEQMSRxtDNNcckAj4kfjXse+Ep+yFOumZtFIZV+eNwn7EKEd/3XosBmcnsrvfiHLAaHp9k+zKxgm/ZRUnWJbFaZAAgnDnURyDpK1XAWiA0rZhgnXRTkflIv5RIblL9xxL+n4qYybi1FeRYG7TW+Gk9g/DUPHx+qbYlbxzuO35Sp5adxELuoUT8Qve2QN2OQuwCkqNK8oPEgqzKderbhW+6wy8Vh+oQB425C/WlCW2T0EEFmZKsYnDAL5HJpGuM5XmEu0RV2zd+Ldg34yn74Mx7QCgSuq757nDf0SSSzN40O00iWFQQgs35Ykwi8LbTgx0cF3PQtsRzO0r3FhRkOSQvwqSbC3TFMvTWzn+MsxPTOfSPRt5vNkQWX40TSMZZH/4vkcYMF44GQ+OXxX7J0E1sUmxu4en9yT0JeDEpkcOdBIQIZpqt3KePrVSGuSgaSoC8n5kMwDBnEWwC1kLsyC/YXjqsm3YJLC+HHs61JaLq18ZL8+7aT3UhaGHzo1D3HIHABXos5mMxSRYTMhz4Ta41frGOEvaxzEo8vW9UPd54KRrHYo5wvOW+iZUl1w73mToJLBUv/ovAnu5RQoln8SzY7VbdEywHnu9nnKpJP44gxzjEeWBfbPc3vb9+mcMuTonExyeSp9N89LA4ZPl0ER+SETHC/rnk8JfzlT3i0o1k8rTYQH5Sko9g+5eM4i/BsA32jjGcUctxgjEerzZOlefe1lSNH3oKVMI+Fm/B80i+uCqt9lCTz8X4T8xsg8Q67foG93X9P9lQPfofsSvxg6Hza/Ok90x8UZLR2Fx8VRpf1a+5h+LjVXLbOs7axc3YS7aKZ+s6pSfgbo9gy8pNtFSflremqm/abVQOqqMjXe/qgsdb7THADr9L9aP6WGXS5X+OitgPxLMVgID7xhNQC+VRy1mIR8tdiJL+IVLAo4BPSeHNORrrvD3sYn8QaS7fUAfPH2R0jZO87yXse+HZ6pZsEueA+6x14shynBH56n/rja+HXXYVC+NPdpz9qIh2Tsqp4TxZOpvm5P+vfSXs++LZis8S7n/w3N4G8jvjQO3oFUU2eDRTvZbe8rYBh034MBX2Odz/BeWecEiP3C2KAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle \\sin{\\left(x y \\right)} + \\cos{\\left(y z \\right)}$"
      ],
      "text/plain": [
       "sin(x⋅y) + cos(y⋅z)"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle y \\cos{\\left(x y \\right)}$"
      ],
      "text/plain": [
       "y⋅cos(x⋅y)"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.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": 88,
   "metadata": {},
   "outputs": [],
   "source": [
    "h = Symbol(\"h\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle y \\cos{\\left(x y \\right)}$"
      ],
      "text/plain": [
       "y⋅cos(x⋅y)"
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.limit((f.subs(x, x+h) - f)/h, h, 0)"
   ]
  },
  {
   "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": 90,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAAKCAYAAABWiWWfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABKklEQVQoFX2S0U3DMBRFHSaIygZhg3aFskEKG8AGVP1K/qqyQdiAlg3aEUhHKBNUYoNwjpNIFKs86eb6Xb9c+9nOuq4LRl3XU2gO7kBL/gYngZ4jPoAZaMEW7RsOmWYkT4wXYA9ugbmxYO7QD+OCJeMVWIJPUIBH8E7dMVRVVYBG0xHkOWhBB+bqcAl2Y81vRt+YZw5wdaWLQLOdLyC76xWarSWB7hFNbvick1kECjwHTYwdSBaMM3yoPUKFZleDIs/rdSh4vlrYT+T/mg0/e8OalpiPFzNM9YTuRZw0O5HYcxLotreG72FbaRhr/jfc9WF8Gg3JksL4XmAPXaM949jmoPmu3IU/b8EEeJbxXUYzkkDxBrJQI03dkbu5CLQXBN+W3Thv3QccfgDHpJb9GsghWgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle \\infty$"
      ],
      "text/plain": [
       "∞"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.limit(1/x, x, 0, dir=\"+\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle -\\infty$"
      ],
      "text/plain": [
       "-∞"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.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": 94,
   "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": 94,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.series(sp.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": 95,
   "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": 95,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.series(sp.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": 96,
   "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": 96,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.series(sp.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": 97,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 1 - \\frac{x^{2}}{2} + \\frac{x^{4}}{24} + O\\left(x^{5}\\right)$"
      ],
      "text/plain": [
       "     2    4        \n",
       "    x    x     ⎛ 5⎞\n",
       "1 - ── + ── + O⎝x ⎠\n",
       "    2    24        "
      ]
     },
     "execution_count": 97,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s1 = sp.cos(x).series(x, 0, 5)\n",
    "s1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAF8AAAAaCAYAAADR2YAqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAERklEQVRoBe2Z4VEbMRCFTYYCCCVAB0AqCHQASQVAB2H4Bf8y0IFJBSR0AKkAQgdQguMOyPtkrdGddZbssy5DnJ0R0kkr7e67p5V8rLy8vPT+y/wInJ+fr2n2qV9hw9eH6h8mVwX8ZS5nZ2f7KrvzYqC5/XAuzypPtb6L8Nna75Jv5x9WEDv3Fd6e6rsWYR5p/m4w/0LtDfVtBX3Xer4Nnl1zacEXGC5dqD6ugzLjM/Mfps2RjUeN/1D9JdRbWvABQ6UfgjFPW4BeqYT5nZfx7AEfL4meHo5V89KdrFrjLdTe8SP5uqliAXPI3frgssKQLnN2VO9lTchU0nqkGlLZdsMUXvY3lQPG3wz4CoxcSm7lJnGJ8yZ6ZkvDqqagTdXqEzVg4sJEtnmh+LitthGjvj42LzS+hk7RtCMDuyowdW7xjv7SAltqExj5syLqg0kccgSfI/g0cQDmTIzpyK4DXjWH91AFX+mrCGPqwP9PDJRmPvmN0kZ+ajJMSbGaGwsHGqxuFIDxg1MPycYFagMeZNLJSbA2eb/JD+xClquizJeBVqJgYDJg5dxInjEWAMBjTEhdHIhNqSE2Z1ofu5I1qa1w/Wxa/0l6OyqvzJcyDLUU8UHtQxW2zmcV5F46N6Nm+b+yhW2Y/Kh2zj3cdth6wjtiawLGTZ0FC+m+T9irD5N22MlrIfM5CC4pGrxX4VQmZ7N9CCw3n0p1IWJsv85cDfYhbgeMmtG/xDKIjrx2lsTCbG848AUwjK+Dy5XJmA6b6uOvrpZpYR9Jsl7+Ayg7hcMuBT6xNDK/AyzM9roduA81p7n/st0tj3JANIr0OHCMeaGeSwEaNxaHY6w/bV3ARFJgomPpMuf6mGJ+KyxwJiHG/FHOFwjkoVAAsum0DvVcuwHcnvphL1fAyr18YoF4B6Az15gS1dI4YJ6qwPocn1mv8VzQGq2wiDpZ7TTbA5d2wjEZNwZbyumpzx0QoV4HbQcCthO2OJvQ+ZjQm3m4EBYWz3DVB8d3Dg4Z8qtLEWqH2/1UzzmsmjnAKRO+aoydww+SaDqRT5xD6PDjps5YdUeFuAyAioLWoL80FhXmw3TKwBsfqD1UcaI+xrj9dCoeTF44pNgKjeOnCr9Q8W1T7eShHMwnlp3gOWx2gYVdDIarsozjMAvDPQXCNxI+VHGI8qOBlzJOQeh0JbLL1Rf/2HlGCmOt7dRZ3WGHuDSqNcck84t0gQW/1B1ZVvivSilRcG0O3FJu9eTXby1+oNqBUMxQZGHZhNB91cU/L8CsOrsiLnXe9V0WF/o5OScCAc6uJYVif3TVpFFC/gazMuMgpfLBrutLBJeHG+HiCDlx1cx0/k2rKXjy/p1q0mKXwssev/ClBN+jzYdDfpx1InrR/Aon14+v8EUP3E6iamHEM5/fCLHPHy1Wrk7V+lwvAb5yziwz83sCgys012p3za5CttAnXu7Ed6w//628ZWbqVKkAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle x + O\\left(x^{2}\\right)$"
      ],
      "text/plain": [
       "     ⎛ 2⎞\n",
       "x + O⎝x ⎠"
      ]
     },
     "execution_count": 98,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s2 = sp.sin(x).series(x, 0, 2)\n",
    "s2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x + O\\left(x^{2}\\right)$"
      ],
      "text/plain": [
       "     ⎛ 2⎞\n",
       "x + O⎝x ⎠"
      ]
     },
     "execution_count": 99,
     "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": 100,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{x^{5}}{24} - \\frac{x^{3}}{2} + x$"
      ],
      "text/plain": [
       " 5    3    \n",
       "x    x     \n",
       "── - ── + x\n",
       "24   2     "
      ]
     },
     "execution_count": 100,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.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": 101,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x - \\frac{2 x^{3}}{3} + \\frac{2 x^{5}}{15} + O\\left(x^{6}\\right)$"
      ],
      "text/plain": [
       "       3      5        \n",
       "    2⋅x    2⋅x     ⎛ 6⎞\n",
       "x - ──── + ──── + O⎝x ⎠\n",
       "     3      15         "
      ]
     },
     "execution_count": 101,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(sp.cos(x)*sp.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": 103,
   "metadata": {},
   "outputs": [],
   "source": [
    "m11, m12, m21, m22 = sp.symbols(\"m11, m12, m21, m22\")\n",
    "b1, b2 = sp.symbols(\"b1, b2\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}m_{11} & m_{12}\\\\m_{21} & m_{22}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡m₁₁  m₁₂⎤\n",
       "⎢        ⎥\n",
       "⎣m₂₁  m₂₂⎦"
      ]
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = sp.Matrix([[m11, m12],[m21, m22]])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}b_{1}\\\\b_{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡b₁⎤\n",
       "⎢  ⎥\n",
       "⎣b₂⎦"
      ]
     },
     "execution_count": 105,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = sp.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": 106,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}m_{11}^{2} + m_{12} m_{21} & m_{11} m_{12} + m_{12} m_{22}\\\\m_{11} m_{21} + m_{21} m_{22} & m_{12} m_{21} + m_{22}^{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡    2                               ⎤\n",
       "⎢ m₁₁  + m₁₂⋅m₂₁    m₁₁⋅m₁₂ + m₁₂⋅m₂₂⎥\n",
       "⎢                                    ⎥\n",
       "⎢                                 2  ⎥\n",
       "⎣m₁₁⋅m₂₁ + m₂₁⋅m₂₂   m₁₂⋅m₂₁ + m₂₂   ⎦"
      ]
     },
     "execution_count": 106,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}b_{1} m_{11} + b_{2} m_{12}\\\\b_{1} m_{21} + b_{2} m_{22}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡b₁⋅m₁₁ + b₂⋅m₁₂⎤\n",
       "⎢               ⎥\n",
       "⎣b₁⋅m₂₁ + b₂⋅m₂₂⎦"
      ]
     },
     "execution_count": 107,
     "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": 108,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle m_{11} m_{22} - m_{12} m_{21}$"
      ],
      "text/plain": [
       "m₁₁⋅m₂₂ - m₁₂⋅m₂₁"
      ]
     },
     "execution_count": 108,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.det()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{m_{22}}{m_{11} m_{22} - m_{12} m_{21}} & - \\frac{m_{12}}{m_{11} m_{22} - m_{12} m_{21}}\\\\- \\frac{m_{21}}{m_{11} m_{22} - m_{12} m_{21}} & \\frac{m_{11}}{m_{11} m_{22} - m_{12} m_{21}}\\end{matrix}\\right]$"
      ],
      "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": 109,
     "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": 110,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ -1, \\  1\\right]$"
      ],
      "text/plain": [
       "[-1, 1]"
      ]
     },
     "execution_count": 110,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.solve(x**2 - 1, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "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": 111,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.solve(x**4 - x**2 - 1, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "方程组："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{ x : 1, \\  y : 0\\right\\}$"
      ],
      "text/plain": [
       "{x: 1, y: 0}"
      ]
     },
     "execution_count": 112,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.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": 113,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$\\displaystyle \\left\\{ x : \\frac{a}{2} + \\frac{c}{2}, \\  y : \\frac{a}{2} - \\frac{c}{2}\\right\\}$"
      ],
      "text/plain": [
       "⎧   a   c     a   c⎫\n",
       "⎨x: ─ + ─, y: ─ - ─⎬\n",
       "⎩   2   2     2   2⎭"
      ]
     },
     "execution_count": 113,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.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."
   ]
  }
 ],
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