{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T13:45:55.592236",
     "start_time": "2017-10-25T13:45:55.028889"
    },
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import sympy as sy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T13:48:35.947972",
     "start_time": "2017-10-25T13:48:35.944558"
    },
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x=sy.Symbol('x')\n",
    "y=sy.Symbol('y')\n",
    "\n",
    "x, y, z = sy.symbols('x y z')     # 符号对象, 可限定 real = True , positive = True, complex = True , integer = True ..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T14:00:16.754202",
     "start_time": "2017-10-25T14:00:16.751464"
    },
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "f = sy.Function(\"f\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T14:16:09.884552",
     "start_time": "2017-10-25T14:16:09.881899"
    },
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "f, g = sy.symbols('f g', cls=sy.Function)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T13:48:25.420147",
     "start_time": "2017-10-25T13:48:25.417418"
    }
   },
   "source": [
    "# 表达式变换\n",
    "\n",
    "- sy.exp(sy.I*x).expand(complex=True) # 展开为复数, 可理解为将x当作复数处理     表达式展开 \n",
    "- sy.cos(x).series(x, 0, 10) #  .series(var, point, order)                 泰勒级数展开   \n",
    "- sy.simplify(expr)          # 对数学表达式进行化简                           化简\n",
    "- sy.radsimp(expr)           # 对表达式的分母进行有理化                       分母进行有理化\n",
    "           它所得到的表达式的分母部分将不含无理数。即将表达式转换为分子除分母的形式。\n",
    "- sy.ratsimp(expr)           # 对表达式的分母进行通分运算                      通分运算 \n",
    "- sy.cancel(expr)           # 对表达式的分子分母进行约分运算                    约分运算\n",
    "           只能对纯符号的分式表达式进行约分\n",
    "           不能对函数内部的表达式进行约分，不能对带函数的表达式进行约分\n",
    "- sy.trim(expr)             # 对表达式的分子分母进行约分运算                   约分运算         \n",
    "           可以对表达式中的任意部分进行约分，并且支持函数表达式的约分运算。\n",
    "- sy.apart(expr)            # 对表达式进行部分分式分解                        部分分式分解\n",
    "- sy.factor(expr)            # 对表达式进行因式分解                           因式分解\n",
    "- sy.together(1/x+1/y)       # 代数式的合并                                 代数式的合并\n",
    "- sy.trigsimp(expr)         # 对表达式中的三角函数进行化简                     三角函数化简 \n",
    "           有两个可选参数——deep和recursive，默认值都为False。\n",
    "              当deep参数为True时，将对表达式中所有子表达式进行化简运算\n",
    "              当recursive参数为True时，将递归使用trigsimp()进行最大限度的化简\n",
    "- sy.expand_trig(expr)      # 展开三角函数                                   三角函数展开 "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 积分"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T13:46:37.227443",
     "start_time": "2017-10-25T13:46:37.202667"
    },
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x**6"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.integrate(6*x**5, x)   "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T13:46:55.153882",
     "start_time": "2017-10-25T13:46:55.145115"
    },
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2500"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.integrate(x**3, (x, 0, 10))          #定积分"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T13:47:03.642073",
     "start_time": "2017-10-25T13:47:03.625472"
    },
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x**6*y + x*y**2/2"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.integrate(6*x**5+y, x,y)             #双重不定积分"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T13:47:13.195048",
     "start_time": "2017-10-25T13:47:13.184671"
    },
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.integrate(x**3+y, (x, -1, 1),(y,1,3) ) #双重定积分"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 微分"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T14:17:23.513604",
     "start_time": "2017-10-25T14:17:23.510116"
    },
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Derivative(f(x), x)\n"
     ]
    }
   ],
   "source": [
    "print f(x).diff(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T13:51:04.744288",
     "start_time": "2017-10-25T13:51:04.739274"
    },
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "cos(x)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.diff(sy.sin(x), x)                                                #    解析微分"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T13:51:19.030395",
     "start_time": "2017-10-25T13:51:19.021344"
    },
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-4*sin(2*x)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.diff(sy.sin(2*x), x, 2) # 高阶微分 diff(func, var, n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T13:51:53.444462",
     "start_time": "2017-10-25T13:51:53.418259"
    },
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x*(x**2*y**2*cos(x*y) + 6*x*y*sin(x*y) - 6*cos(x*y))"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.diff(sy.sin(x*y),x,2,y,3) # 对表达式进行x的2阶求导,对y进行3阶求导"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T13:52:03.023314",
     "start_time": "2017-10-25T13:52:03.018842"
    },
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2*cos(2*x)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.diff(sy.sin(2*x),x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T13:52:16.120448",
     "start_time": "2017-10-25T13:52:16.115922"
    },
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2*cos(2*x)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.sin(2*x).diff(x,1)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T13:52:37.507763",
     "start_time": "2017-10-25T13:52:37.504503"
    },
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "t=sy.Derivative(sy.sin(x),x)  \n",
    "# Derivative是表示导函数的类，它的第一个参数是需要进行求导的数学函数，第二个参数是求导的自变量，\n",
    "# 注意：Derivative所得到的是一个导函数，它不会进行求导运算。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 极限"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T13:53:01.658125",
     "start_time": "2017-10-25T13:53:01.584689"
    },
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.limit(sy.sin(x)/x, x, 0)       # 极限"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 求和"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T13:53:48.524201",
     "start_time": "2017-10-25T13:53:48.486937"
    },
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "y**2"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.summation(2*x - 1, (x, 1, y))     #  连加求和"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 替换"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T13:54:22.015575",
     "start_time": "2017-10-25T13:54:22.003114"
    }
   },
   "source": [
    "expression.subs(x, y)   #          将算式中的x替换成y     替换\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T13:54:43.157915",
     "start_time": "2017-10-25T13:54:43.152946"
    },
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "pi*y + 1"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(1 + x*y).subs(x, sy.pi)  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "expression.subs({x:y,u:v})        使用字典进行多次替换  \n",
    "\n",
    "expression.subs([(x,y),(u,v)])    使用列表进行多次替换   "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T13:55:14.735400",
     "start_time": "2017-10-25T13:55:14.729637"
    },
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1 + 2*pi"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(1 + x*y).subs([(x, sy.pi), (y, 2)])\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 方程\n",
    "\n",
    "方程构建: 任何表达式都可以直接表示值为0的方程,即表达式 expr 也是方程 expr=0\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T13:58:59.391335",
     "start_time": "2017-10-25T13:58:59.377274"
    },
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# 用sy.Eq构建方程\n",
    "u_max, rho_max, A, B = sy.symbols('u_max rho_max A B')     # 符号对象, 可限定 real = True , positive = True, complex = True , integer = True ...\n",
    "eq1 = sy.Eq( 0, u_max*rho_max*(1 - A*rho_max-B*rho_max**2) )  # 一般方程"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T14:16:20.655852",
     "start_time": "2017-10-25T14:16:20.650434"
    },
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Eq(f(x)**2 + f(x) + Derivative(f(x), x), 0)\n"
     ]
    }
   ],
   "source": [
    "eq1 = sy.Eq(f(x).diff(x)+f(x)+f(x)**2,0)   # 微分方程\n",
    "print eq1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T14:00:29.649982",
     "start_time": "2017-10-25T14:00:29.646593"
    }
   },
   "source": [
    "sy.solve(方程,未知数)   # 方程求解 ,返回值为list     方程求解\n",
    "\n",
    "                    它的第一个参数是表示方程的表达式，其后的参数是表示方程中未知量的符号。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T14:16:30.278154",
     "start_time": "2017-10-25T14:16:30.144042"
    },
    "collapsed": false
   },
   "outputs": [
    {
     "ename": "NotImplementedError",
     "evalue": "\nNo algorithms are implemented to solve equation _Dummy_172 + f(x)**2 + f(x)",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mNotImplementedError\u001b[0m                       Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-53-3a39fcc28893>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0msy\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msolve\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0meq1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;32m/Users/chengjun/anaconda/lib/python2.7/site-packages/sympy/solvers/solvers.pyc\u001b[0m in \u001b[0;36msolve\u001b[0;34m(f, *symbols, **flags)\u001b[0m\n\u001b[1;32m   1051\u001b[0m     \u001b[0;31m###########################################################################\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1052\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mbare_f\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1053\u001b[0;31m         \u001b[0msolution\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_solve\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0msymbols\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mflags\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1054\u001b[0m     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1055\u001b[0m         \u001b[0msolution\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_solve_system\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msymbols\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mflags\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/chengjun/anaconda/lib/python2.7/site-packages/sympy/solvers/solvers.pyc\u001b[0m in \u001b[0;36m_solve\u001b[0;34m(f, *symbols, **flags)\u001b[0m\n\u001b[1;32m   1617\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1618\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mresult\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0mFalse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1619\u001b[0;31m         \u001b[0;32mraise\u001b[0m \u001b[0mNotImplementedError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'\\n'\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mjoin\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mmsg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnot_impl_msg\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1620\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1621\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mflags\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'simplify'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mTrue\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mNotImplementedError\u001b[0m: \nNo algorithms are implemented to solve equation _Dummy_172 + f(x)**2 + f(x)"
     ]
    }
   ],
   "source": [
    "sy.solve(eq1,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T14:13:30.862972",
     "start_time": "2017-10-25T14:13:30.845496"
    },
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ConditionSet(x, Eq(x + sin(x), 0), Complexes((-oo, oo) x (-oo, oo), False))"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.solveset(sy.sin(x)+x,x) # 即对 sy.sin(x)+x=0 求解"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "sy.roots(expr)     # 计算单变量方程expr=0的根                                   方程求解"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T14:18:07.319953",
     "start_time": "2017-10-25T14:18:06.709571"
    },
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Eq(f(x), -C1/(C1 - exp(x)))"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.dsolve(eq1,f(x))  # 微分方程求解                                            微分方程求解"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "可以对微分方程进行符号求解，它的第一个参数是一个带未知函数的表达式，第二个参数是需要进行求解的未知函数。\n",
    "\n",
    "它在解微分方程中可以传递hint参数，指定微分方程的解法，若设置为“best”则放dsolve()尝试所有的方法并返回最简单的解。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T14:19:17.259701",
     "start_time": "2017-10-25T14:19:17.256983"
    },
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sympy.utilities.lambdify import lambdify"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T14:19:47.136254",
     "start_time": "2017-10-25T14:19:46.949978"
    },
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<function numpy.<lambda>>"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    " lambdify([x],f(x),modules=\"numpy\") # 第一个参数为自变量列表"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T14:20:08.439390",
     "start_time": "2017-10-25T14:20:08.434866"
    },
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.1415926535897932385"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.N(sy.pi,20)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T14:20:18.335649",
     "start_time": "2017-10-25T14:20:18.331422"
    },
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.1415926535897932384626433832795028841971693993751"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.pi.evalf(n=50)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T14:21:07.071614",
     "start_time": "2017-10-25T14:21:07.069201"
    },
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "i=sy.Symbol('i')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T14:21:11.609961",
     "start_time": "2017-10-25T14:21:11.604452"
    },
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Sum(cos(pi*i)*x[i], (i, 1, 10))"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = sy.Sum(sy.Indexed('x',i)*sy.cos(i*sy.pi),(i,1,10))  \n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T14:21:38.297145",
     "start_time": "2017-10-25T14:21:38.291485"
    },
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "<function <lambda> at 0x108494b90>\n"
     ]
    }
   ],
   "source": [
    "func = lambdify(x, f, 'numpy') # returns a numpy-ready function\n",
    "print func"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-10-25T14:21:52.955199",
     "start_time": "2017-10-25T14:21:52.881170"
    },
    "collapsed": false
   },
   "outputs": [
    {
     "ename": "TypeError",
     "evalue": "'Symbol' object does not support indexing",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-68-adb2d1657b9e>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mnumpy_array_of_results\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m      2\u001b[0m \u001b[0mnumpy_array_of_results\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/chengjun/anaconda/lib/python2.7/site-packages/numpy/__init__.pyc\u001b[0m in \u001b[0;36m<lambda>\u001b[0;34m(_Dummy_204)\u001b[0m\n",
      "\u001b[0;32m/Users/chengjun/anaconda/lib/python2.7/site-packages/numpy/__init__.pyc\u001b[0m in \u001b[0;36m<genexpr>\u001b[0;34m((i,))\u001b[0m\n",
      "\u001b[0;31mTypeError\u001b[0m: 'Symbol' object does not support indexing"
     ]
    }
   ],
   "source": [
    "numpy_array_of_results = func(y)\n",
    "numpy_array_of_results "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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