2017-12-15_Sympy_101/2017-12-15_Sympy_101.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Sympy 101"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import (init_printing, S, sqrt, I, pi, exp,\n",
    "                   Symbol, symbols, Eq, solve, limit,\n",
    "                   Limit, oo, Integral)\n",
    "from sympy.stats import Normal, E"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "init_printing(use_latex=\"png\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'0.333333333333333314829616256247'"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f\"%.30f\" % (1/3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAqBAMAAACXcryGAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAVO8Qq4lmdpnNIrvdRFil\nQ4MAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAB1SURBVBgZY2CAASEDIIvZ1Q9EMTDkUUbVgbW36M1I\nAJtGFvEOBB4QpXVp6EagOq4GBrkEBgb2hwx8BxgYeHUZ+CaAdYMEgV47CiK5YppBFAND0QUwxaQM\nFFvAwP2QgYHvMZhi2sDA85iBgS0BbBjDjY69DAwAM4MfYQyrCzEAAAAASUVORK5CYII=\n",
      "text/plain": [
       "1/3"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S(1)/3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABwAAAAvBAMAAAACzbekAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAInarRM2ZVBDdiWbv\nuzJCz3LGAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+ElEQVQoFWNggAL5/yAA4zGYwFkgBmMACpet\nAIXbicJjWM7AuGotXAXHBAYxBvbfMCVcGxj0Cxh+wrhArT4ODF9hXG0QA6yYxQHImgDidi4AEjpA\nBzApABlS0xqAzmn6wsDACeQBBY8ACY6/DAybwFyGMw1AOl+AYQaQEmRguA9yh3wC9wIg9b8BwmX+\nwdkA5J5lYJgPovl+lQFJht0wR8ZEgLgsabkOIJqhPgFMwQheARiLljQ4LuAE1W1Cjh+g4cjxA+Qi\nxw+Qixw/YJcgIhPMBccPmAUiwPED50HjB4kPjh8oHxY/UC4sfqBcWPxAufD4gfDh8QMApQdLgo68\nk/cAAAAASUVORK5CYII=\n",
      "text/plain": [
       "√3\n",
       "──\n",
       "3 "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sqrt(S(1) / 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiXZmMs1UEN0i77ur\nRJlR0qN3AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAUUlEQVQIHWNgYFQWYWBgCGOomMDAvICBMYCB\n+wAD23cG/gMMvN8Y6h8w8H5imC/AwAIkHzCwfISKAGXZvjFwb2Bg/g7VxdDGUOXAwFCodIQBAG3H\nFgUteuAKAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sqrt(S(1) / 3) - sqrt(3) / 3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABgAAAAPBAMAAAAMihLoAAAAJFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAADHJj5lAAAAC3RSTlMAEM0yVO+riWZ2mSMU5s8AAAAJcEhZcwAA\nDsQAAA7EAZUrDhsAAAAsSURBVAgdY2DAClgCEcJiZRsRHAYOMjlCSiCgzECWAZxIlnKlblqB5B4G\nBgDVtwt2YFScIgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "-1"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp(I * pi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADoAAAAWBAMAAABj+5JWAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAdu+ZiTJUIma7qxDd\nRM2CxnTmAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAvElEQVQoFWNgwAmcJ2/AKcfAUsD1ALcsTwHr\ndzyyF9j/4pZlYGD7jU+WRwCfrDM+SQ4DVNlEFO4VhqsofBRZ1mdps3HLMv3//4GBQej///+fIYpg\nehFCLK6JFeIFKLJIQmEM7owBUPOhepGFGCZwbQDJ+r17d+jdu1dghVAhIJt1AQ9YBEjA7EUSYm7g\nB+tFkkUS2r+B/wJUM0wvkpAMA5MDmiyS0DQGLnRZJCGoPhAFMxlJCIm5G4kNZgIAPZU0F3csf+8A\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "     2\n",
       "π + π "
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pi + pi ** 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAE8AAAAUBAMAAAA+U32BAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAdu+ZiTJUIma7qxDd\nRM2CxnTmAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABHUlEQVQoFWNggAP2BjgTnZGOIrAThYfC4SpA\n5j4Hcti0kUXgbO4HcCYDA4cBA8Mur99QETYBJCkGBkckHg/IeA4cCjchKawBsXEp5LvAwCD0////\nz0A1wlgVQmVZDBhYXBMrxEG2zsCmECbL8YAhjMGdMQCk5hg2hTBZtgNA2QlcG0BqdEEExI0s7969\n0Xv3bgFIBCLL+oGBgXUBD0gAWSGQBwseqCxIIXMDP9hErFbDZHkPMDDs38B/AWQiVs/AZIGeYZBh\nYHJgAAJnEMGIHuBQWWYHBoZpDFxAkoHhDhAzL/pnAWIj3AiV5QuACINIcBTCuTDPQAX2wCUgiQLB\n5QZ7EM5HThQM0+HCGAyUZMaAJ+FyFiDrxZMV2hkYAB+vRvEyj903AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "π⋅(1 + π)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "obj = (pi + pi ** 2).factor()\n",
    "obj"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sympy.core.mul.Mul"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "obj.func"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGcAAAAUBAMAAABi2T6lAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/plain": [
       "(π, 1 + π)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "obj.args"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABIAAAAQBAMAAADpKDJvAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mSJE3e9UMqtm\nzbsXyEShAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAZUlEQVQIHWNgAAMhYwMIg/Ei4wYIi3vPGQiD\ngSMBymDgD2AQgLD5JsDUMTobA4UYlU0CIDLpAawKYBZjJ4PIAjCLu8f4IJjBwA81nIGBXwEiwsDA\nDmQZgDnMFxhEoKaYnDkAFgIAekYOshTqbGgAAAAASUVORK5CYII=\n",
      "text/plain": [
       " ℯ\n",
       "ℯ "
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp(exp(1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x = Symbol(\"x\", real=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAJBAMAAAAWSsseAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAEHarIkSJZt3NuzJUmW69\n3xMAAAAJcEhZcwAADsQAAA7EAZUrDhsAAABASURBVAgdY2AQUnZVU2BgTGBv4pjAwCbA9pDVgYGR\ngWsBAwjwKYCpfRuAFI+AHgOTAEPcgXUM7gwMwkpC1wsYABfXCcn8wW65AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "x"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAC8AAAAQBAMAAAB0JTvnAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NuzJU\nme9mZ+xlAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAuklEQVQYGWNgwASMd1dtwBQFiggzsH1FkeBR\ngHAvMjC8wyrxgoHB3gFZBqZjvgCDPdASIWVXNQWwPEwCyOkXYGBMYG/imIAmwfqNgYFNgO0jqwOa\nBIcBAwMjA9cCkDDbzJkzXs6cCRQBgrtgkl8BTDEwwO1gugAW2b8BXeIsA4sDA4+APgOTAFgKpoM7\ngYHJgSH+wHoGd4gWmIRHWq4FMGCUhK4XoErM////B0QESsJ0oAiCOIwQqyDiANlmIkuIM1hRAAAA\nAElFTkSuQmCC\n",
      "text/plain": [
       "x + 2"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x + 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGAAAAAQBAMAAAAblGfKAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NuzJU\nme9mZ+xlAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABMElEQVQoFWNgwASMd1dtQBPFIoSkQpiB7SsS\nF8TEIgQU5VEAyTEwXGRgeAdhwUksQkA5mIYXDAz2DnC1YAYWISQN8wUY7NE8gSwkpOyqpgA2BmYD\nkNMvABZBJvoFGJVMgpIYGBPYmzgmoGlg/QYSyJwJAnMhuoBCYgwVCp0MbAJsH1kdwIIIGzgMIKqQ\nSKBQIkO+QAgDIwPXApA428yZM17OnAlReRdJJZQJFBJgWAXm8CtAxeA2MF2AiiAoiNAnsMD+Dega\nzjKwOADFIH6YDJYFC3H/BrJ5BPQZmATAgjAbuBMYmBzAInACLHSQ6QODIkP8gfUM7hAJmAaPtFwL\nuFIIAyTE+ofrA0sBg7CS0PUCVA3z////gaYBJMRopKhbjiIOswFFEB+HEeIVfEoYGADe5kfbw4cH\nogAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "x + 2 = 2⋅x"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq(x + 2, 2 * x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABIAAAAUBAMAAAByuXB5AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAds0yiUTdEJm7q+8i\nVGZXpJ2YAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAc0lEQVQIHWMQUmYAgUepDK4MDBWdkxmAtCsD\n+wIG/wAwi6OAgXsDmMWdwMD1B8xi+s7A9RnMAupk+w5jxTvAWFkQUxgYOBpgLAmgUpDJDLwNDGcg\nLBMGBhkwiyfFpSsBzGL6//8/hAVUDtOBhyUGcb1QKgCVzhyh68CyMwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "[2]"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(Eq(x + 2, 2 * x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "a, b, c, d = symbols(\"a b c d\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAfsAAAAVBAMAAACznFqnAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nDSYwVimduaCfAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "⎡                                                      π⎤\n",
       "⎣2⋅a + b - c = 12⋅d, π⋅a - b - c = ℯ, a + 15⋅b + 37 = ℯ ⎦"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "system = [\n",
    "    Eq(2 * a + b - c, 12 * d),\n",
    "    Eq(pi * a - b - c, exp(1)),\n",
    "    Eq(a + 15 * b + 37, exp(pi)),\n",
    "]\n",
    "system"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAABPwAAAAyBAMAAAA6vtgSAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "⎧              π                                      ⎛   π     ⎞             \n",
       "⎪   180⋅d - 2⋅ℯ  - 15⋅ℯ + 74     -12⋅d + ℯ - (-π + 2)⋅⎝- ℯ  + 37⎠     12⋅d⋅(1 \n",
       "⎨a: ────────────────────────, b: ────────────────────────────────, c: ────────\n",
       "⎪          -15⋅π + 28                       -15⋅π + 28                        \n",
       "⎩                                                                             \n",
       "\n",
       "                         ⎛   π     ⎞⎫\n",
       "+ 15⋅π) - 29⋅ℯ + (2 + π)⋅⎝- ℯ  + 37⎠⎪\n",
       "────────────────────────────────────⎬\n",
       "         -15⋅π + 28                 ⎪\n",
       "                                    ⎭"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solution = solve(system, [a, b, c])\n",
    "solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMkAAAAtBAMAAAD2PVKiAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nvMq8zVnt2NvEnI+1bX7otZFXDqN6jxCQhY4AAAAASUVORK5CYII=\n",
      "text/plain": [
       "           π            \n",
       "180⋅d - 2⋅ℯ  - 15⋅ℯ + 74\n",
       "────────────────────────\n",
       "       -15⋅π + 28       "
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solution[a]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAX8AAAAQBAMAAAAMv2JvAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYZGA+8F1Tzgc7tQ9s/8A8+kR1+huegIAAAAASUVORK5CYII=\n",
      "text/plain": [
       "-9.41231108557171⋅d + 0.682686044855686"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solution[a].evalf()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAK8AAAAPBAMAAABgoIKoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/plain": [
       "-347.572824121298"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solution[a].evalf(subs={d: 37})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHYAAAAQBAMAAAAxJJZlAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRK5CYII=\n",
      "text/plain": [
       "-c + 2 = 12⋅d"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "system[0].subs({a: 0, b: 2})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEQAAAAPBAMAAACvonQuAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAzWYQMplU74mrdiK7\nRN1/7zyFAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABFElEQVQoFXWPsUoDQRRFzyYxZnddiYWFVUR7\nmU62S69gokTsYmNlJSIWEVkLwdgYCxEsTGWjjRDS+gEimk/Qzk4lENLoOm+2GZd4i5k79x0ed3Dm\nc+uMULb2VJZ4eY/xOI7E2srvQkjh22QDnOadPRUfNvuwBvvyCPq4cqeU0cghtO91Xnj9Fykp2mWN\nOBXcuRtxfyRbtD4U3dOVTSaUb2odTYmmzSxBgiFBgxcl0bPJrSNBJlsszlI1eV1q2UqQGtQV73AO\nZz17rr1BvAockNUtdgwLSZcZAxtkC78zJP8Jmn1LLTFbchFeZ4Bb3KZLppFGPP3ph5Pjax5ZaEX4\nGxcqhYxdfq1SiuMfwtulq15qOur5C3B0Q0BFqyEhAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "5 = 12⋅d"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "system[0].subs({a: 0, b: 2, -c + 2: 5})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFgAAAA0BAMAAAD8jeImAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\noO3Vf/EUhSIk14oUAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "       x\n",
       "⎛    1⎞ \n",
       "⎜1 + ─⎟ \n",
       "⎝    x⎠ "
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "likee = (1 + 1 / x) ** x\n",
    "likee"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAALBAMAAABv+6sJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKvvZom7mXYyzVQi\n3UQ6SGZXAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAbUlEQVQIHWNgYBBgAAIQwaj82YGBIayogYGB\nbQLHLwapDQwnGBg8GRj2J6xkYGA5wACUYP0LJBgcQEyGfAcg0QBmJv8DandgcGRg4IgxPsDAcYCB\nnYEhnGGyHYMrUKJcbQMDY71xAFA5BABtAwBKehUH3PvDFwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "∞"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "oo"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIoAAAA0BAMAAACnab1SAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAKm+AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "           x\n",
       "    ⎛    1⎞ \n",
       "lim ⎜1 + ─⎟ \n",
       "x─→∞⎝    x⎠ "
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Limit(likee, x, oo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAJBAMAAAASvxsjAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAEIl2mSJE3e9UMqtmzXCQ\nkgMAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAA4SURBVAgdY2BgVDYJYGBID2BVYGDsZBBZwMDdY3yQ\ngYFvAwMQ8CmASHYgacDAfIFBBKjS5MwBBgDXogiIdN7mwgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "ℯ"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Limit(likee, x, oo).doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAJBAMAAAASvxsjAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAEIl2mSJE3e9UMqtmzXCQ\nkgMAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAA4SURBVAgdY2BgVDYJYGBID2BVYGDsZBBZwMDdY3yQ\ngYFvAwMQ8CmASHYgacDAfIFBBKjS5MwBBgDXogiIdN7mwgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "ℯ"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(likee, x, oo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEYAAABKBAMAAAD0wj/MAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nTdEfu9Mkt/oNmeKGkaRtlNYAAAAASUVORK5CYII=\n",
      "text/plain": [
       "5      \n",
       "⌠      \n",
       "⎮  2   \n",
       "⎮ x  dx\n",
       "⌡      \n",
       "0      "
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Integral(x ** 2, (x, 0, 5))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACAAAAAqBAMAAAAgz0ShAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq4lmdpnNuzJE\n3SJtjddMAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABB0lEQVQoFWNgAAOLrtUMDNMMdgiAeEIGDMwO\nDPEHGPT/fwVymV39DRg4DBiYFBimuj8ACjAw5AN5FxhY/jE4gLlgAc7PDCy/kQWAUryfGdzSu8Fq\ngFqA4HwAw2KG9wdATIjAFRCT6wJcgKMBxOT4BxeoYGBgmcDA+gcmwNjA8I5pAgPHZ5jAVgaGOg6g\nVRNAAvUGDHxXQ1su8DUwJD0A8lv0ZyRw/v///wLDi5BmkIJBAoBOQgG0dBZfaBQ4WuF2vGBg/Abn\ngBj1Agx/UQQOHWD4iSLAgK6FgcE2AFXF0xsGqAIM7DpoAgxKKEreMTCsR3HZfwM0AS0GhvsoWtLQ\n3cHT0XUAYgsAayxSPy2s9soAAAAASUVORK5CYII=\n",
      "text/plain": [
       "125/3"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "5 ** 3 / S(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACAAAAAqBAMAAAAgz0ShAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq4lmdpnNuzJE\n3SJtjddMAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABB0lEQVQoFWNgAAOLrtUMDNMMdgiAeEIGDMwO\nDPEHGPT/fwVymV39DRg4DBiYFBimuj8ACjAw5AN5FxhY/jE4gLlgAc7PDCy/kQWAUryfGdzSu8Fq\ngFqA4HwAw2KG9wdATIjAFRCT6wJcgKMBxOT4BxeoYGBgmcDA+gcmwNjA8I5pAgPHZ5jAVgaGOg6g\nVRNAAvUGDHxXQ1su8DUwJD0A8lv0ZyRw/v///wLDi5BmkIJBAoBOQgG0dBZfaBQ4WuF2vGBg/Abn\ngBj1Agx/UQQOHWD4iSLAgK6FgcE2AFXF0xsGqAIM7DpoAgxKKEreMTCsR3HZfwM0AS0GhvsoWtLQ\n3cHT0XUAYgsAayxSPy2s9soAAAAASUVORK5CYII=\n",
      "text/plain": [
       "125/3"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Integral(x ** 2, (x, 0, 5)).doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAOgAAAAWBAMAAAA4H80iAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/plain": [
       " 5    4    3    2        \n",
       "x  + x  + x  + x  + x + 1"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "func = x ** 5 + x ** 4 + x ** 3 + x **2 + x + 1\n",
    "func"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAOMAAAAWBAMAAADA/zbVAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSNFQC51R15feg7LcUBnKe90g6oxq/x+6fMo9ch5K9AAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "   4      3      2          \n",
       "5⋅x  + 4⋅x  + 3⋅x  + 2⋅x + 1"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "func.diff(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANMAAAAcBAMAAAAJl5eLAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJm7MquJRO/dIs12\nVGbfGimAAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADeUlEQVRIDbVVTUhUURT+3jjOOOPM+FoYRYFm\nZItCTHHtrGqpUUhF4WTkpqFmEREROJJQ0aJwFVY0tChqkRK1iOlnFJSgwKlNLQpdiW10jErKajrn\nvXvvu8/xKcF0F/d+5zs/3z333jcDrDICY3u9vT0tprfznz1N2OqZE8hUjng613LsKQloxHwJJ4lQ\nxv9T4rXXiCvE3+kybcO7q1AqvLhCggcVdBWvXuHojQGPVKajC6s4l7tO6sRO3RD4dW4FUlKh1Zwy\nSK4hasRoaI2zHVZ7TM5YBJOlzyI5oZw9VoQzhXuP5R1rGYrUA7UIWGceIIxoP1CdN4ZU3PCy5E0p\n34hwxjhBH3cQ/aPbCn+gjsIFYAtwickqKlp7nFqbBU7YUXRTw9M2/GgvaEFFs4BncNpG0Zy9Tubx\nQ/isRfBGVzdJ4RZwGejOEBxmO0ZSbcAUYxpDmBVISAWd9+1//mmXHSWlzqawZDNuHvNchN7ySxPd\ncYLUjC31DejLswHsnjhkAwipirSw6RSKxYJtSCk6f9cBKt6SemUF32bVHQypK+MLSeWMN4fOJy2n\nPQmpmsfJ7cDm8XsTOcepSmJjAlqi4i2pPhYJfue0Vp5IKkxWR249PuduMCOGkKp7gVDeSMRuhkak\nh3rJCXx30ISWqHhLqiNOUaFODn3CE0tRVx2JozhlnmNGDClVgK8/YAYWghnp0aQQuQgt0S01nKKM\nBitrkGd1gKbdpeXBiWz2ejbLe6m5htii4bxCIJDNPr2SzVq7pbdsykSdt7qqIyn+uGg84omkQM9i\nKg/8YlsN0RWdW4yuvianHATk7h8A7Xqi5O0XyF29QzhDCW2czVJ0abQN/1e21RBSlWnqig44rhwE\nZMmiyVJOouRtqY4M/AlEMpSwj7NZyvqE30YKGGVGDCHlX4AvHTUnETGlx5G6ANw2tUS3FL3A+709\nBznNevcRkqInNhD8XVEITzvV5HeFLmzI9+Xb8VDzyZJz9F3piZLHJO+LOnhZLFo/J/zufQeWmmAk\nx+PG4dGxOa2akgo07kftm81N05pTlgw2bE3piZLfdvURhWv/fFV0bd5DHOCKAbLkcqebf+a4fQkH\nl6J1pZRiDO3eFEnAxRt0NXL40xL9lzXWqZU9ouHyw1Bcq7lJw+WHM3rJaL1ulRmHm10F37us8ho+\nj6dTXhVV7S/uL/KEkW19wgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "  ⎛    3      2          ⎞\n",
       "2⋅⎝10⋅x  + 6⋅x  + 3⋅x + 1⎠"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "func.diff(x, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "mu = Symbol(\"mu\", real=True)\n",
    "sigma = Symbol(\"sigma\", positive=True)\n",
    "dist1 = Normal(\"x\", mu, sigma)\n",
    "dist2 = Normal(\"x\", -mu, sigma)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiXZmMs1UEN0i77ur\nRJlR0qN3AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAUUlEQVQIHWNgYFQWYWBgCGOomMDAvICBMYCB\n+wAD23cG/gMMvN8Y6h8w8H5imC/AwAIkHzCwfISKAGXZvjFwb2Bg/g7VxdDGUOXAwFCodIQBAG3H\nFgUteuAKAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E(dist1 + dist2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFcAAAAYBAMAAAB96HzmAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJm7MquJRO/dIs12\nVGbfGimAAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABs0lEQVQ4EXWSv0vDQBzFX2qbNGmqQXA2VhRB\nwZ+TUyZRHOwqDi0dFEWwgwhOOioqZlFBBDM4qEvFTYq0CDqIQ9G5VPwLrIiiUM5rk0vvWrzl3vu8\nx/G9SwBupUYMzolS7u4SgOyErgTAmyXM8RaqE/wVAG+OMePwXk1r37wX9BEyaQEgWha94EoNF1Lz\nQiyaE0CKjVo+TPnKFe3dYz6RTaADsj9oK/X8kuJY8Qd9owl9vS1WWMOqK6P52h4wEN50CZSkYmEb\nSDguCN4Wh13llcM2Quwxp4svBnIGEpZbCRPyLpQjZYQ+XYJ9Qmrq1MA5cO/R6uadTJVahtRHyAcL\nlS9gEThkXihvJDG1sD7usFCNA9eQKswL5QNIe2j1pqNJjD52hft4cjZ7s5PN0iOAgInAFRT/w+om\nnbFcZfXlz3wHtL1Ct1n0DM0J2FCTBiP1C0ZNXGYsREwvCiahO+oQMoV8c3kCeOw0MMnOuZhPzaLT\nxuBZoamsHc732C0F7YglOUJ+UOp90pcZobs3c4QQYksDD+xgt9HPFavSv2ADr9ndBiiJZwmpxn4A\ngf5jFPufoAH/AZkSZ5lfrDyzAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "   2      2\n",
       "2⋅μ  + 2⋅σ "
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E(dist1 ** 2 + dist2 ** 2)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
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   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
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 },
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