LieTransformations.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import sympy\n",
    "from operator import mul\n",
    "from functools import reduce\n",
    "from mpmath import fac\n",
    "import functools\n",
    "from sympy import init_printing\n",
    "from IPython.display import display_latex\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Poisson bracket\n",
    "$$:f: = \\left[f, \\cdot \\right]\n",
    "= \\frac{\\partial f}{\\partial x} \\frac{\\partial \\cdot }{\\partial p} - \\frac{\\partial f}{\\partial p} \\frac{\\partial \\cdot}{\\partial x} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "class explieoperator:\n",
    "    def __init__(self,ham, indep_coords, indep_mom)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 161,
   "metadata": {},
   "outputs": [],
   "source": [
    "class LieBracket:\n",
    "    \"\"\"\n",
    "    Lie bracket class\n",
    "    \"\"\"\n",
    "    def __init__(self, ham, indep_coords, indep_mom):\n",
    "        self._ham = ham\n",
    "        if not isinstance(indep_coords, list):\n",
    "            indep_coords = [indep_coords]\n",
    "        if not isinstance(indep_mom, list):\n",
    "            indep_mom = [indep_mom]\n",
    "\n",
    "        assert(len(indep_coords)==len(indep_mom))\n",
    "        self._indep_coords = indep_coords\n",
    "        self._indep_mom = indep_mom\n",
    "        \n",
    "    @property\n",
    "    def ham(self):\n",
    "        return self._ham\n",
    "    \n",
    "    @ham.setter\n",
    "    def ham(self, other):\n",
    "        self._ham = self._ham.subs(self.ham, other).doit()\n",
    "        \n",
    "    @property\n",
    "    def indep_coords(self):\n",
    "        return self._indep_coords\n",
    "    \n",
    "    @indep_coords.setter\n",
    "    def indep_coords(self, other):\n",
    "        self._indep_coords = other\n",
    "        \n",
    "    @property\n",
    "    def indep_mom(self):\n",
    "        return self._indep_mom\n",
    "    \n",
    "    @indep_mom.setter\n",
    "    def indep_mom(self, other):\n",
    "        self._indep_mom = other\n",
    "        \n",
    "    def ApplyPoissonBracket(self, other):\n",
    "        summ = 0\n",
    "        for i in range(len(self.indep_coords)):\n",
    "            summ += self.ham.diff(self.indep_coords[i]) * other.diff(self.indep_mom[i]) \n",
    "            summ -= self.ham.diff(self.indep_mom[i]) * other.diff(self.indep_coords[i])\n",
    "\n",
    "        return summ\n",
    "    \n",
    "    def PowApplyPoissonBracket(self, other, power):\n",
    "        summ = 0\n",
    "        if power == 0:\n",
    "            summ += other\n",
    "        else:\n",
    "            summ += self.ApplyPoissonBracket(self.PowApplyPoissonBracket(other, power - 1))\n",
    "            \n",
    "        return summ\n",
    "    \n",
    "    def p(self,n):\n",
    "        F = sympy.Matrix(np.zeros((n+1,n+1)))\n",
    "        for i in range(n+1):\n",
    "            for j in range(n+1):\n",
    "                if (j-i)>=0:\n",
    "        #             F[i,j] = 1/(np.math.factorial(j-i))\n",
    "                    F[i,j] = sympy.Rational(1,int(fac(j-i))) # keep it rational !!!!\n",
    "\n",
    "\n",
    "\n",
    "        G = sympy.Matrix(np.zeros((n+1,n+1)))\n",
    "        for i in range(n+1):\n",
    "            for j in range(n+1):\n",
    "                if (j-i)>=0:\n",
    "                    if i==j:\n",
    "                        sigprod = 1\n",
    "                    else:\n",
    "                        l = sympy.symbols('s{}:{}'.format(i,j))\n",
    "                        sigprod = reduce(sympy.Mul,l)\n",
    "                    G[i,j] = sympy.Mul(sympy.Rational(1,int(np.math.factorial(j-i))),sigprod, evaluate=False)\n",
    "\n",
    "\n",
    "\n",
    "        qthpower = sympy.Matrix(np.eye(n+1))\n",
    "\n",
    "        FGm1 = sympy.nsimplify(F*G - qthpower)\n",
    "\n",
    "        FGm1\n",
    "\n",
    "\n",
    "        summ = []\n",
    "\n",
    "        for q in range(1,n+1):\n",
    "            qthpower = sympy.nsimplify(sympy.MatMul(qthpower, FGm1, evaluate=False))\n",
    "\n",
    "            summ.append(sympy.nsimplify((-1)**(q-1)/q * qthpower)[0,n]) # q-1 because mathematica starts at 1 and not 0, there for sign change\n",
    "\n",
    "        return sympy.expand(reduce(functools.partial(sympy.Add,evaluate=False),summ))\n",
    "\n",
    "    def translated(self,n,ham1_symbol,ham2_symbol):\n",
    "        l = sympy.symbols('s{}:{}'.format(0,n))\n",
    "        temp = reduce(functools.partial(sympy.Mul,evaluate=False),[k*k for k in l])\n",
    "        temp = sympy.expand(sympy.Mul(temp,  self.p(n), evaluate=False),evaluate=False)\n",
    "        with sympy.evaluate(False):\n",
    "            ln = sympy.symbols('r{}:{}'.format(0,n), commutative=False)\n",
    "            for i, s in enumerate(l):\n",
    "                temp = temp.subs(l[i],ln[i])\n",
    "            for i in range(n):\n",
    "                g = sympy.Symbol('g',commutative=False)\n",
    "                f = sympy.Symbol('f',commutative=False)\n",
    "                temp = temp.subs(ln[i]**3,g)\n",
    "                temp = temp.subs(ln[i]**2,f)\n",
    "        \n",
    "        temp = temp.subs(f,ham1_symbol)\n",
    "        temp = temp.subs(g,ham2_symbol)\n",
    "\n",
    "        return temp\n",
    "    \n",
    "    def SumTranslated(self,n,ham1_symbol,ham2_symbol):\n",
    "#         summ = sympy.expr()\n",
    "#         for i in range(1,n+1):\n",
    "#             summ = sympy.Add(summ,self.translated(i,ham1_symbol,ham2_symbol))\n",
    "            \n",
    "        return reduce(functools.partial(sympy.Mul,evaluate=False),[self.translated(i,ham1_symbol,ham2_symbol) for i in range(1,n+1)])\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 220,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAwAAAAUBAMAAABL3sEiAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIma7zZnddlTvRImrEDIioekeAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAYklEQVQIHWNgYFRgEmBgYBBmYDgIpBwZGK4wMDAFMDBwFzCwASV4JzCIAcUZvBhKQdRWBiMQpcKQAaLaGGaBqPcwCiy4DKZkK0gummEZiPIDG8aYADOawRBsEchaIIsB7AgAXCcPpfvdQgkAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left ( \\right )$$"
      ],
      "text/plain": [
       "()"
      ]
     },
     "execution_count": 220,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x.args"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 162,
   "metadata": {},
   "outputs": [],
   "source": [
    "x,p = sympy.symbols('x p')\n",
    "H = sympy.Function('H', commutative=False)(x,p)\n",
    "H2 = sympy.Function('H2', commutative=False)(x,p)\n",
    "g = sympy.Function('g', commutative=False)(x,p)\n",
    "la = LieBracket(H,x,p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 163,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\left ( H{\\left (x,p \\right )}, \\quad \\left [ x\\right ], \\quad \\left [ p\\right ]\\right )$$"
      ],
      "text/plain": [
       "(H(x, p), [x], [p])"
      ]
     },
     "execution_count": 163,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "la.ham, la.indep_coords, la.indep_mom"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 164,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$- \\frac{\\partial}{\\partial p} H{\\left (x,p \\right )} \\frac{\\partial}{\\partial x} g{\\left (x,p \\right )} + \\frac{\\partial}{\\partial x} H{\\left (x,p \\right )} \\frac{\\partial}{\\partial p} g{\\left (x,p \\right )}$$"
      ],
      "text/plain": [
       "  ∂           ∂             ∂           ∂          \n",
       "- ──(H(x, p))⋅──(g(x, p)) + ──(H(x, p))⋅──(g(x, p))\n",
       "  ∂p          ∂x            ∂x          ∂p         "
      ]
     },
     "execution_count": 164,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "la.ApplyPoissonBracket(g)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 222,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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ho9royAx+eT1yajxlfycsfiV/anx1l07ydMIUYMJojzy2juwvbuBlT9vYwZO6q+vn4SgN4lvTeEbXAD/HZkxjhydHi06NM9EK2RALB60TopLdxSn57rQaHVKnxpWSesZCFaGUY01URlRozpAxdok59ihROF1Ownoqlnns8CSsCp+PtKBK7Ey1R+upSojEqXFWT3WFb1Ic7urLnugagpU3lneXzsDJMe7Dn6rA55Efv8CVoH7K/jW4bwPgH7vmbHtwHV6cdsMUYLJoo+mJMQa8F/8u0SHpEfznfPUS3mQ5oE+viYxOTk9cz5nGDk+OFkbgohWyQRJS/SrZnd8g351Wo8PXOhizqZXUMxaqiJ8l8hNdE5UFFZozZIxdYo49ShLOlJOwnoplHjs8uYLqvApLYmNiMmiJnan2aD1VCXFsy4vte7yeyin6n5y/450/mBy/9NoEX2anYfDcRx/8RUz4p9Hv5S89OMV3HBuAjf3Kesp+5fVz/4SWbyfz4KfjfSj4ywc+3g0TwBQ2gAujDad47n3w3ZvPwTdv/hE6wo+ad5y79e/PANyNR6tERivPffJ0ztQenqIiYAQmWpANS5wJIswIKtmd3TbieZZVOsANg1MrD1ZKgp6xUMUgRwv2uVZF9XGE5hjPh4qx43NsyTGILJwWQYbBoKUyt4cnlom0KlhKUmiBnan2SD3VvbBgbeZ0ZwcOXk/hgSZST/GfnPfXmtO1BXgkwfYptYz/ynpgjx/6sS2pJXfP+0Anedi9HBiGw1QoXIQQO1oTbbi9RB9TXRtObXcVe8mM8IwYuzeia2oPT1ERfAToZMOz4H3LKCNEaF7F7tq28Vtg2U0ufKKzBzslccZ8joSun7EQl1TfClJnHmrTY8ejWMchxI5WhHMiCDDKNF/m5vAUp5V+eXB73rfkqgsqy85Ue76eOiURiB48wNsVlJMyG9yk8iGA+3xWtne/7cAF7N2ptj5By8UxLFxRm8EisMcP/dhGE1py96y/AAM6eCRhBKXGIHpALZOwwBwvE2JtcWY3RthJZLQBJ7f5naUdUxhMlZO4CD4CiVYhgvKlF1UZVbLbnGifJZbd5IJbggMwKalnzOdIaJ6jjpmasRCXUN+6yMxAQIvNdowdi+I8M4gbqwwXlhO7/bijI6dYLPPbNIs4rfTLg9vzvs+ppqBK7Ey15+spLQR+EzXxhLgsvp7SKRrkt/BEzBnvxfSWp6YzmOBHljdoYzDGBRzHY9Mr1AlbYH/yKu28RVlw96x/cX15B/cnYQqLCwaxQzlYYD6cMAjA5+3WrcmM8OIufKRf2tR6SIjgIpBoFSJYd7hOCsEyqmW3NtOOSyw7OsCqmmZDKgAPJ/gdg54xlyOigxwNLjFjNepbD7TmYZLlx7SJsOMw55pB3FhluOGEI+oo7q3Mq6oiIXhNQVWyK9RTcq4A+M3pvKBIysrgqPlwhouwuadELeL40y9PcOkSjv3KOrdfvffHyZmrMO7e9v/rw/9CNjkY7admIXpLLXMwb/4IQ2D3M3YTD8aJjFb/50F1ZM2ZWi/32I5fO9gifktYIYJHZoWwGdWyM/ftKO85ll0dRtcZIeo6MClpZszliOhujhZuCdttWnNcQv0Wcxsqwq4bxbq3ELtN6xp2JAJvNZi9lXlVVRClSEo1L5FKdoV6yojX/c7PFhSrpxxPpXb8J+c/zWdC9T+iRxK/st6zX5xqe+6e9/XeJEzvhsQvzydhLIL/PKF94deQnZbMKGlqHcRF6MPqRCCvNRmZ6P0wJhGz//iu6dSyNPATFqfgDMyV7Ae3OZqgiRnr45Lqk6M6czbbOnoSZsm1s+MiKG/JWEw5GzY5u9agl4nekcRxe963/kQFlYyivRXqKT1X9npz7cbLEkhZCK5++Fzf3O1zS/f0r6z7W//TlnqPcm/u4xeGIgdCiDI3DyZ4HL5YIqf264zqFUCQFLIHETi0Kh08k2LOuSmW5qECNdDga2A+z7VKqiBCwkKSmtYe68k/VqIkC2NXLQJXruTf7heKpl8WbSqoUKJqD+qJPZPDkk+v+ZUqLVLid/t4QdOlib65Ox3H7/nyBiy/ym7993uiPe1eP/xAGgodCiHaXLNbeMezn41S6g6qjKBeAcQLIXsQgUO7xFPbC+YckmJpHiqQsuXjj07YFpvnaiUJLiUsJKmV32M9sWdxsIxjXc+uXgSmXMxlbEwqGrfX/eryFZoT26Ce2DM5YpkEY/Sb5661SGl++Pwv9CMJnKt0R//Kur/1P22p91DJ37utH36g+vWhyIEQoqNpdsfUPcuaRW6pMzJ3WucM/T4pZA8icKgnUOg9qfZrluahAgWE2n1+6q34PNcqqdBCwkKS2tw8F0KFaqgnU44+2WSPsasWgSuXdNzZIRRNvyzaXlUNogGvJ6N9J4H45tHrfrxFSkS/uA7wdXNzt3eW7OlfWfe3/icNzQ7l3tzHLwxFHoQQHa2eHYWwGamHMtBAuUkhexCBQ8vEjMVTaq1ZmocK1GA/x4xsike22GBNV0hYSFKb77Ge2GMlChkJ2SlvLcoJRdMvizYVVKj617tKiddTvXj4Zdppr2+LlIh+Gb9l/765gd87S/bWxvh18gW8FmGSNAl2aPf64QfCUORHCLHRatlRCJORvr2dBspNCrG06L1uY0b8bXKR4OVtMjEsL6unRxQxWAY8hnSenX+eqxtMd4QkLS31XAihlERif9mpNC1FSQ0KaQVpCFUQmquUgnpiDzJROzOLkxO/UzjRBmh++Bwv/8VHElQ0sjqPb7g2NyqMseRv4IOB6Zo+fPiBNBQGEEJstFp2KgWTUbUCCBJCLK0GETi0SnBlNJzRyrDUDxVQ4/kFzZNrwnl2OClhIUlLq0FKorjP7JQKlmLlK6SFFk/D9GvLV2iuMjLXLdjE/DM59N70kg4Utgkn2sDMD58fUTfwW1eZNR0Q38Nv/c/Y4i7jXt3HLw3l4bXsTAT2YII8O7VXZ0QXQONDGeqaELIHETi0jhtZ6a9aDEv9UIEK8LEdZiScZ4eUEhaSNOZ7rCf2WAlHPNoRslM+GpSTisbtTb+2fIXmWpWgnqrFA3POSvtokRKReJX08/hBQD2SQDvKLpd34MSWufU/a2h2KvfmPn5hKPIghGhz/WCCGnZkozOqV6ABsgcROLQ2I4C/YYnp+9wrsNemzEg6zw4qJGzUryWpzfdYTwbuKKc7QnbKUYtyQtH0y6JNBRVKVO2YFaunevFglZ0Zty+z2om2c6J/+Fzf3G3HMmv9K+vm1v+Mnd2l3Jv7+IWhyIUQos3r2VEI87vx1Qo0QPYgAocS27qGj3G2iZmbwCtwd3Ib6Tw7rJCwjlNNUpvvsZ4M3FFOd4TslKMW5YSi6ZdFmwoqlHlgRzrxzh5WT/XimTsyjasWKQmqfvhc32fcIRXdZL+yHt3fHeS/qy4MRa6EEB6tyyS1rTOqVwD9SCGc1hwyMoWhWOJJibq2+pXATjrPDsxzdYPpjoykUd64E0pJqP1lp3i1KCekFaQhVEForlLCj47YpIk9E5wsEU60imsXwX9OO7g/64ZQDRAx94YYDRBLaw9Q6yK3XnKPYQ7/heUwt2/n9u7vPl3+bTH2WUoktRd2bTnJUUIVhOasnuqpDZ7o27ZJOeD3qvedvpkjDaEaIGLGDTEaIJbWHqDWRX79tNutH5zgNtOdx9K79n1PNck+k32X0j6XoR/6AI0IVRCaF34TLqEDPZus29omWt/c3fW1L9sNoRogYuoNMRogltYeoNZFfn18x+w3DxXIW9Pexa2yzX5ZVJOMENh3Ke1zGSKxD86QUAWhOT5GaUee6y/1IXuZ6L63w5EfWgWekzL/ghRwaP9WUkBcT7Ayfivpc5irSIHj2yJzGHxHZn9o/dZSQFpP+GD/4MS4TK3/B6T2WWz099xrAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$- \\frac{\\partial}{\\partial p} H{\\left (x,p \\right )} \\frac{\\partial}{\\partial x} H{\\left (x,p \\right )} \\frac{\\partial^{2}}{\\partial x\\partial p} g{\\left (x,p \\right )} - \\frac{\\partial}{\\partial p} H{\\left (x,p \\right )} \\frac{\\partial^{2}}{\\partial x^{2}} H{\\left (x,p \\right )} \\frac{\\partial}{\\partial p} g{\\left (x,p \\right )} + \\frac{\\partial}{\\partial p} H{\\left (x,p \\right )} \\frac{\\partial^{2}}{\\partial x\\partial p} H{\\left (x,p \\right )} \\frac{\\partial}{\\partial x} g{\\left (x,p \\right )} + \\left(\\frac{\\partial}{\\partial p} H{\\left (x,p \\right )}\\right)^{2} \\frac{\\partial^{2}}{\\partial x^{2}} g{\\left (x,p \\right )} - \\frac{\\partial}{\\partial x} H{\\left (x,p \\right )} \\frac{\\partial}{\\partial p} H{\\left (x,p \\right )} \\frac{\\partial^{2}}{\\partial x\\partial p} g{\\left (x,p \\right )} - \\frac{\\partial}{\\partial x} H{\\left (x,p \\right )} \\frac{\\partial^{2}}{\\partial p^{2}} H{\\left (x,p \\right )} \\frac{\\partial}{\\partial x} g{\\left (x,p \\right )} + \\frac{\\partial}{\\partial x} H{\\left (x,p \\right )} \\frac{\\partial^{2}}{\\partial x\\partial p} H{\\left (x,p \\right )} \\frac{\\partial}{\\partial p} g{\\left (x,p \\right )} + \\left(\\frac{\\partial}{\\partial x} H{\\left (x,p \\right )}\\right)^{2} \\frac{\\partial^{2}}{\\partial p^{2}} g{\\left (x,p \\right )}$$"
      ],
      "text/plain": [
       "                             2                           2                    \n",
       "  ∂           ∂             ∂              ∂            ∂           ∂         \n",
       "- ──(H(x, p))⋅──(H(x, p))⋅─────(g(x, p)) - ──(H(x, p))⋅───(H(x, p))⋅──(g(x, p)\n",
       "  ∂p          ∂x          ∂x ∂p            ∂p            2          ∂p        \n",
       "                                                       ∂x                     \n",
       "\n",
       "                   2                                      2   2               \n",
       "    ∂             ∂            ∂             ⎛∂          ⎞   ∂             ∂  \n",
       ") + ──(H(x, p))⋅─────(H(x, p))⋅──(g(x, p)) + ⎜──(H(x, p))⎟ ⋅───(g(x, p)) - ──(\n",
       "    ∂p          ∂x ∂p          ∂x            ⎝∂p         ⎠    2            ∂x \n",
       "                                                            ∂x                \n",
       "\n",
       "                        2                           2                         \n",
       "         ∂             ∂              ∂            ∂           ∂             ∂\n",
       "H(x, p))⋅──(H(x, p))⋅─────(g(x, p)) - ──(H(x, p))⋅───(H(x, p))⋅──(g(x, p)) + ─\n",
       "         ∂p          ∂x ∂p            ∂x            2          ∂x            ∂\n",
       "                                                  ∂p                          \n",
       "\n",
       "              2                                      2   2         \n",
       "             ∂            ∂             ⎛∂          ⎞   ∂          \n",
       "─(H(x, p))⋅─────(H(x, p))⋅──(g(x, p)) + ⎜──(H(x, p))⎟ ⋅───(g(x, p))\n",
       "x          ∂x ∂p          ∂p            ⎝∂x         ⎠    2         \n",
       "                                                       ∂p          "
      ]
     },
     "execution_count": 222,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "la.PowApplyPoissonBracket(g,2).expand()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 175,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADUAAAARBAMAAACP9fljAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiXZmMs1Eu5kiEO9Uq90itUqIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAqElEQVQYGWNgIAOcXoCh6RBEhOvGfZxyDAz6hOQYlUyQTIaaCdUXxlDBwCQkAJVHkeP8wODFYMhwFpsc1+82BoZnDPwgq0VDQ1+HhsaClYHdkv6/gOsLA/cBsAgDiplzGBb9ZPnJwGeARS6ageEDyy8GPgUscpUMTApAfdzY5CYp6zAA7eNPQNXXuz8WagvUnUBpuFsgSoGkBtx/K+FiMAZjYxeMiUQDABkUMge9IrhBAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$s_{0} + 1$$"
      ],
      "text/plain": [
       "s₀ + 1"
      ]
     },
     "execution_count": 175,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "la.p(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 176,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- \\frac{s_{0}}{2} + \\frac{s_{1}}{2}$$"
      ],
      "text/plain": [
       "  s₀   s₁\n",
       "- ── + ──\n",
       "  2    2 "
      ]
     },
     "execution_count": 176,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "la.p(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 166,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{s_{0} s_{1}}{24} - \\frac{s_{0} s_{2}}{12} + \\frac{7 s_{0}}{12} + \\frac{s_{1} s_{2}}{12} - \\frac{s_{1} s_{3}}{12} + \\frac{s_{1}}{3} + \\frac{s_{2} s_{3}}{24} + \\frac{s_{2}}{12} + 1$$"
      ],
      "text/plain": [
       "s₀⋅s₁   s₀⋅s₂   7⋅s₀   s₁⋅s₂   s₁⋅s₃   s₁   s₂⋅s₃   s₂    \n",
       "───── - ───── + ──── + ───── - ───── + ── + ───── + ── + 1\n",
       "  24      12     12      12      12    3      24    12    "
      ]
     },
     "execution_count": 166,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "la.p(1)+la.p(2)+la.p(3)+la.p(4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 167,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$H{\\left (x,p \\right )} + \\operatorname{H_{2}}{\\left (x,p \\right )}$$"
      ],
      "text/plain": [
       "H(x, p) + H₂(x, p)"
      ]
     },
     "execution_count": 167,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "la.translated(1,H,H2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 168,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAgMAAAAsBAMAAAAKkKauAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZqvNmRDdRHYyiVS7Iu/EmopNAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAFfklEQVRoBc1YX4gbRRz+klySZnO51D/4Vg0V5V6Kx+XwDwoN5fRBKiz2TdAL/kepd09S+3KLXOFexII++GQC1lKkcnmxCFXMg4I+1ItwCCJyB9IHnwxXr7XeSfzN/kl2dmd2J7sbk4GZ3Zn5Zn7ffDc3O1+A6Okn39AjvhbW4MOlOkJc3MbQOKEARQZ39hZAWUep7RuRZmsrPdWbpfy90yvAHXP64jy3lneNreUDw5lDFieIcEQiNyjmTcoZ/98yO8f4FA5IB8p2EuBOOX2xno02sE3ZTtI4AYSjEcn/TYv8i8Le7cR2PS+x98IuJ4EApxmuMZFfGxVOAlmcIMLRiGi3gUP7xPu0gPsLrM0jgQCX3RGMHbrJI4EsThDhaEQyXWB6j/7OdQHnzAo18hIIcY8Ixg7dxEsgjRNIOBKRchso12kn1IDs+rEz5t4Hzl776F0dGjV6JOBx+cu/rD8AXCBY7MRLII0jJhyHyOr80aObNdoJh4HncI/xkrWU5oMtrY0UOw8LNxcXT/SPQx6npd/EZeCu2OunCRovLy4+1HZmksYRE45DZOM88H4LyFH+FBv61yaHUmseqS6Kdao5/wjaySepxuN+zVRwH3DFHBSzcHbBhW/1oDhiwnGIHKdwmxRzZgXQ8YS9jCz2oe2h0KW6I8HnuJ9qPE4vt0g2NOxhsR62BFonPRcUR0w4DhG2w9mXli0NoC+klfJ7mN7hJTiNRseHW9XxI+1hZ1icpy1BrlP4l6bx8BnEkRAeAIblYH5l/6FRbIOjQF9IK2l1LNWQr1PN2QWvYZtk8uAuIkuXirftUbEejgQrpVs0jyyOjHB0IuZXloVkx88Xh7o4Zy0jU8O9un0c9q9G3+g+3AJS15M6Div21ahI32gvn34cGeE+wOI/RJnrUjQWkr5/+f3pbukwluaoWm4X/gCmmvTKLsgsA69T9uAO8PF54GnWGzexCzLLyBlUyOLICIuJKDipZ4/vGlObzJvQns/Onlu7gxi8QgxWz6x1yDfQ8mybNGuy8uLyt9afIfRjlAOTAhXLJu0a9i2D59OPIyPcBzAaUZ3U4ELKvosX2VT42Sydwrq1czg6Mij5bnKqrs+ZmXumamY1NE4AIKKTGtiSJlFYMGlcMku7yDfztClcdqpJ26TGOjWDle7ENvQ2ZTtJXZ8DcD/P4h1W5fiI4sgBSk7qS3dM632aNr6ZsgZdlZlxgmWWzUYqPrj6lU4PDrfUYr0Dj1o0WJ3+pyucBAO2Zi8r/LLZXYXHr37HXgPjBAOUnJRAgv5PFCma/8oNw82CRcSfvR57uHHF5UdZ0zVWmEkiwWDPOkBITd1Mr9dlqMA4wQAlJyWQwP+DGI70Cbtfgo42sQS+s4JNF2rqguKYfKSAcltk/TxOSiSBOW3MQiyB1PXFjCYbruSkRi6BmuuTrSFmu5KTGrkEFes4DHF9MZcqGx7upD6pVt+qVudlE0Ru16rVh3+vVis0gaLrw6lXWQq9WQ1JSclJObugl1QyD3HAcxaEuT730hJiQkTUnJQjgZtDEu9eCYJdXxIRBXOoOan/SQLaFkGuT0A/iSY1JzVyCdoqri+J9frnUHJSwIglUHN9fvZJtwxupWT9PJZuxBL0l6Lq+voDkn0ZeJOmz9I9rx5q6+QP6uBsh8equj5+lKQ2FBNzjlCrJYnEN2cr2Fjhm9Rryq5PZcoITEKtlkpcpHXM7CghBSBl1ycY62uKwkTqpHyzBzTM1JG+HdCv1pUElWSYqPHlUJm9JCTgpoxYGSeTHLveTEQaG5PV5kSsn0iMjcmLk6IAxsVkqjYpEoyNydqkKIBxMSnW8NtkiDA2Jh8C702GBONiUnrjsxP1iZBgbEwy9IPWZEigyOQ/a60FUiVUVo4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$H{\\left (x,p \\right )} + \\frac{H{\\left (x,p \\right )} \\operatorname{H_{2}}{\\left (x,p \\right )}}{2} + \\operatorname{H_{2}}{\\left (x,p \\right )} - \\frac{\\operatorname{H_{2}}{\\left (x,p \\right )} H{\\left (x,p \\right )}}{2}$$"
      ],
      "text/plain": [
       "          H(x, p)⋅H₂(x, p)              H₂(x, p)⋅H(x, p)\n",
       "H(x, p) + ──────────────── + H₂(x, p) - ────────────────\n",
       "                 2                             2        "
      ]
     },
     "execution_count": 168,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "la.translated(1,H,H2)+la.translated(2,H,H2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 169,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKUAAAAUBAMAAAAek+A5AAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZqvNmRDdRHYyiVS7Iu/EmopNAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAC9ElEQVQ4Ea1VTWgTQRT+tmmSZpPY+IM3NRQUL8XSFFRUGqEnQVjqpQjagD8IpbYnqb10KS0EL+3Rkw2oxUOhuSiCCCvosSaFUhEFexHxFlpjq43U783ObtYkRx/Mzvfe++bNz3szCwRkNYAVNMqNlgY9mWow4IgyHNjrA5uFpNNIQHeDZW1s014bq9me+bMH/L7NXcYWDT/ZQs2ruuJzNVhwgC9sWi54wO/DPQKjO0DsB8Eh3+ED09bwpe4X0sGYRlabA90zweYvoGOXYDTg0TC8oUHrmJHmneG6jAhVgEQVSOb0+GB3WiutY64HqRqHxgk6HbYc15oFwvnuCbV4RJc+5o8DT0gQaR1zhp6h6Q8PbKFgcuXhtAUzSzjV29VVIkikgKs4bN+gjSfSNoIl4KBSAjFvDgyccjjXW4ueN1xFwbgT6VGswomi6cAQZXkemCsCEbanWLZeK8KnUBrHgBdKCcRMqxyZ5TYZOsK5LbMaLwsrWeyFUUE8R9zPVrKAfeOAhfPURKzOIuNjQfDjTOZuJtMrUOc9Uo79ofaN60TIEYegXZhVxCrENTYpUYkJ7MhHZMrCe8ZwlcbzjIwnt+lhTObCli8lWkViQ8VU5fmbJtk7YiwsVxYRZtHe01pTjuKsFNk7p53XHDOH4SyiOV2eMqnk6HlHBbMupQ/G1xY5Suuaj9ikMUdxaw4dlhoSyuKopXIUqbjlKUUQ3U1UkikM95BTw6N54JJiB3LkcGFsbpGx0qaKJawiLIfX6cS+A+0FDPZv2u2lms3jyCF8cnZmP0/hFrXt/GXyzrKJ6L27b8imDfdW8mQG80MXU8Bti+EnZsq8QlxJXep3k+XEw6H4d8s7T489ifuE9bsZZ7RF5Vz3KKqvvyEFTpcVm2nLl3LN7bxv7Nyrd8T1N8Sk1qe87jX0iAnOpSRsA8NFgU1vnfKz7vb2KgJXtI4h1qe8Q3DfOs9cf5MNpnPsjNj9MT7pX+C/ySleuS2bTn9hmvif/h1/Ab970I0O2upOAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$H{\\left (x,p \\right )} + \\operatorname{H_{2}}{\\left (x,p \\right )}$$"
      ],
      "text/plain": [
       "H(x, p) + H₂(x, p)"
      ]
     },
     "execution_count": 169,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "la.SumTranslated(1,H,H2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 174,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left(H{\\left (x,p \\right )} + \\operatorname{H_{2}}{\\left (x,p \\right )}\\right) \\left(\\frac{H{\\left (x,p \\right )} \\operatorname{H_{2}}{\\left (x,p \\right )}}{2} - \\frac{\\operatorname{H_{2}}{\\left (x,p \\right )} H{\\left (x,p \\right )}}{2}\\right) + \\operatorname{H_{2}}{\\left (x,p \\right )}$$"
      ],
      "text/plain": [
       "                     ⎛H(x, p)⋅H₂(x, p)   H₂(x, p)⋅H(x, p)⎞           \n",
       "(H(x, p) + H₂(x, p))⋅⎜──────────────── - ────────────────⎟ + H₂(x, p)\n",
       "                     ⎝       2                  2        ⎠           "
      ]
     },
     "execution_count": 174,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "la.SumTranslated(1,la.SumTranslated(2,H,H2),H2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$H{\\left (x,p \\right )} + \\frac{H{\\left (x,p \\right )} \\operatorname{H_{2}}{\\left (x,p \\right )}}{2} + \\operatorname{H_{2}}{\\left (x,p \\right )} - \\frac{\\operatorname{H_{2}}{\\left (x,p \\right )} H{\\left (x,p \\right )}}{2}$$"
      ],
      "text/plain": [
       "          H(x, p)⋅H₂(x, p)              H₂(x, p)⋅H(x, p)\n",
       "H(x, p) + ──────────────── + H₂(x, p) - ────────────────\n",
       "                 2                             2        "
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "la.SumTranslated(2,H,H2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$H{\\left (x,p \\right )} + \\frac{H{\\left (x,p \\right )} \\operatorname{H_{2}}{\\left (x,p \\right )}}{2} - \\frac{H{\\left (x,p \\right )} \\operatorname{H_{2}}{\\left (x,p \\right )} H{\\left (x,p \\right )}}{6} + \\frac{H{\\left (x,p \\right )} \\operatorname{H_{2}}^{2}{\\left (x,p \\right )}}{12} + \\frac{H^{2}{\\left (x,p \\right )} \\operatorname{H_{2}}{\\left (x,p \\right )}}{12} + \\operatorname{H_{2}}{\\left (x,p \\right )} - \\frac{\\operatorname{H_{2}}{\\left (x,p \\right )} H{\\left (x,p \\right )}}{2} - \\frac{\\operatorname{H_{2}}{\\left (x,p \\right )} H{\\left (x,p \\right )} \\operatorname{H_{2}}{\\left (x,p \\right )}}{6} + \\frac{\\operatorname{H_{2}}{\\left (x,p \\right )} H^{2}{\\left (x,p \\right )}}{12} + \\frac{\\operatorname{H_{2}}^{2}{\\left (x,p \\right )} H{\\left (x,p \\right )}}{12}$$"
      ],
      "text/plain": [
       "                                                                  2          2\n",
       "          H(x, p)⋅H₂(x, p)   H(x, p)⋅H₂(x, p)⋅H(x, p)   H(x, p)⋅H₂ (x, p)   H \n",
       "H(x, p) + ──────────────── - ──────────────────────── + ───────────────── + ──\n",
       "                 2                      6                       12            \n",
       "\n",
       "                                                                              \n",
       "(x, p)⋅H₂(x, p)              H₂(x, p)⋅H(x, p)   H₂(x, p)⋅H(x, p)⋅H₂(x, p)   H₂\n",
       "─────────────── + H₂(x, p) - ──────────────── - ───────────────────────── + ──\n",
       "      12                            2                       6                 \n",
       "\n",
       "        2           2              \n",
       "(x, p)⋅H (x, p)   H₂ (x, p)⋅H(x, p)\n",
       "─────────────── + ─────────────────\n",
       "      12                  12       "
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "la.SumTranslated(3,H,H2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "def operatorpow(n,op,g,x,p):\n",
    "    return PoissonOperator()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$e^{H{\\left (x,p \\right )} + \\operatorname{H_{2}}{\\left (x,p \\right )}}$$"
      ],
      "text/plain": [
       " H(x, p) + H₂(x, p)\n",
       "ℯ                  "
      ]
     },
     "execution_count": 106,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "esp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {},
   "outputs": [
    {
     "ename": "AttributeError",
     "evalue": "'Mul' object has no attribute 'exp'",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mAttributeError\u001b[0m                            Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-117-c31c867286ca>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mexp\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mAttributeError\u001b[0m: 'Mul' object has no attribute 'exp'"
     ]
    }
   ],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "la.ham = x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$3 p^{2}$$"
      ],
      "text/plain": [
       "   2\n",
       "3⋅p "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "la.ApplyPoissonBracket(g).subs(g,p**3+x).doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 187,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy.physics.quantum.operator import HermitianOperator, Operator"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 188,
   "metadata": {},
   "outputs": [],
   "source": [
    "A = Operator('A')\n",
    "B = Operator('B')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 216,
   "metadata": {},
   "outputs": [
    {
     "ename": "TypeError",
     "evalue": "zip argument #2 must support iteration",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-216-4515ca00fb8c>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m      2\u001b[0m \u001b[0mfu\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msympy\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mFunction\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'fu'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      3\u001b[0m \u001b[0mexpr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msympy\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mexp\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mu\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mseries\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mu\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;36m3\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 4\u001b[0;31m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msubs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mu\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mfu\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\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      5\u001b[0m \u001b[0;31m# expr.subs(u,la.SumTranslated(2,H,H2))\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/core/basic.py\u001b[0m in \u001b[0;36msubs\u001b[0;34m(self, *args, **kwargs)\u001b[0m\n\u001b[1;32m    928\u001b[0m             \u001b[0mrv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    929\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0mold\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnew\u001b[0m \u001b[0;32min\u001b[0m \u001b[0msequence\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 930\u001b[0;31m                 \u001b[0mrv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mrv\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_subs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mold\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnew\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    931\u001b[0m                 \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mrv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mBasic\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    932\u001b[0m                     \u001b[0;32mbreak\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/core/basic.py\u001b[0m in \u001b[0;36m_subs\u001b[0;34m(self, old, new, **hints)\u001b[0m\n\u001b[1;32m   1042\u001b[0m         \u001b[0mrv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_eval_subs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mold\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnew\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1043\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mrv\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1044\u001b[0;31m             \u001b[0mrv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfallback\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mold\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnew\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1045\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mrv\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1046\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/core/basic.py\u001b[0m in \u001b[0;36mfallback\u001b[0;34m(self, old, new)\u001b[0m\n\u001b[1;32m   1014\u001b[0m                 \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mhasattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'_eval_subs'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1015\u001b[0m                     \u001b[0;32mcontinue\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1016\u001b[0;31m                 \u001b[0marg\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0marg\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_subs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mold\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnew\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1017\u001b[0m                 \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0m_aresame\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\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[1;32m   1018\u001b[0m                     \u001b[0mhit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;32mTrue\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/core/basic.py\u001b[0m in \u001b[0;36m_subs\u001b[0;34m(self, old, new, **hints)\u001b[0m\n\u001b[1;32m   1040\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mnew\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1041\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1042\u001b[0;31m         \u001b[0mrv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_eval_subs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mold\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnew\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1043\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mrv\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1044\u001b[0m             \u001b[0mrv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfallback\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mold\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnew\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/series/order.py\u001b[0m in \u001b[0;36m_eval_subs\u001b[0;34m(self, old, new)\u001b[0m\n\u001b[1;32m    443\u001b[0m                             \u001b[0me2\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msol\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    444\u001b[0m                             \u001b[0msol\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mset\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0me1\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0mset\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0me2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 445\u001b[0;31m                         \u001b[0mres\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mdict\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mzip\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0md\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msol\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    446\u001b[0m                         \u001b[0mpoint\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0md\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msubs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mres\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[0mlimit\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mold\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpoint\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    447\u001b[0m                     \u001b[0mnewvars\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mvar\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mTypeError\u001b[0m: zip argument #2 must support iteration"
     ]
    }
   ],
   "source": [
    "u = sympy.Symbol('u')\n",
    "fu = sympy.Function('fu')\n",
    "expr = sympy.exp(u).series(u,n=3)\n",
    "expr.subs(u,fu(x))\n",
    "# expr.subs(u,la.SumTranslated(2,H,H2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 178,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = sympy.Function('a', commutative=False)\n",
    "b = sympy.Function('b', commutative=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 186,
   "metadata": {},
   "outputs": [
    {
     "ename": "PolynomialError",
     "evalue": "non-commutative expressions are not supported",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mPolynomialError\u001b[0m                           Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-186-f296df2035f4>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0msympy\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mexp\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mseries\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36mseries\u001b[0;34m(self, x, x0, n, dir, logx)\u001b[0m\n\u001b[1;32m   2664\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0ms\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msubs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mxpos\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ms\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrv\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2665\u001b[0m             \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2666\u001b[0;31m                 \u001b[0;32mreturn\u001b[0m \u001b[0mrv\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msubs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mxpos\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[1;32m   2667\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2668\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mn\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m  \u001b[0;31m# nseries handling\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/core/basic.py\u001b[0m in \u001b[0;36msubs\u001b[0;34m(self, *args, **kwargs)\u001b[0m\n\u001b[1;32m    928\u001b[0m             \u001b[0mrv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    929\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0mold\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnew\u001b[0m \u001b[0;32min\u001b[0m \u001b[0msequence\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 930\u001b[0;31m                 \u001b[0mrv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mrv\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_subs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mold\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnew\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    931\u001b[0m                 \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mrv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mBasic\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    932\u001b[0m                     \u001b[0;32mbreak\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/core/basic.py\u001b[0m in \u001b[0;36m_subs\u001b[0;34m(self, old, new, **hints)\u001b[0m\n\u001b[1;32m   1042\u001b[0m         \u001b[0mrv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_eval_subs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mold\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnew\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1043\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mrv\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1044\u001b[0;31m             \u001b[0mrv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfallback\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mold\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnew\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1045\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mrv\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1046\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/core/basic.py\u001b[0m in \u001b[0;36mfallback\u001b[0;34m(self, old, new)\u001b[0m\n\u001b[1;32m   1014\u001b[0m                 \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mhasattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'_eval_subs'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1015\u001b[0m                     \u001b[0;32mcontinue\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1016\u001b[0;31m                 \u001b[0marg\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0marg\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_subs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mold\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnew\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1017\u001b[0m                 \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0m_aresame\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\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[1;32m   1018\u001b[0m                     \u001b[0mhit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;32mTrue\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/core/basic.py\u001b[0m in \u001b[0;36m_subs\u001b[0;34m(self, old, new, **hints)\u001b[0m\n\u001b[1;32m   1040\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mnew\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1041\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1042\u001b[0;31m         \u001b[0mrv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_eval_subs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mold\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnew\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1043\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mrv\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1044\u001b[0m             \u001b[0mrv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfallback\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mold\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnew\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/series/order.py\u001b[0m in \u001b[0;36m_eval_subs\u001b[0;34m(self, old, new)\u001b[0m\n\u001b[1;32m    438\u001b[0m                         \u001b[0;32mfrom\u001b[0m \u001b[0msympy\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msolvers\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msolveset\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0msolveset\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    439\u001b[0m                         \u001b[0md\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mDummy\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 440\u001b[0;31m                         \u001b[0msol\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msolveset\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mold\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0mnew\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msubs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvar\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0md\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0md\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    441\u001b[0m                         \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msol\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mComplement\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    442\u001b[0m                             \u001b[0me1\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msol\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/solvers/solveset.py\u001b[0m in \u001b[0;36msolveset\u001b[0;34m(f, symbol, domain)\u001b[0m\n\u001b[1;32m   1143\u001b[0m     \u001b[0mf\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mpiecewise_fold\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1144\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1145\u001b[0;31m     \u001b[0;32mreturn\u001b[0m \u001b[0m_solveset\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msymbol\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdomain\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0m_check\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mTrue\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1146\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1147\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/solvers/solveset.py\u001b[0m in \u001b[0;36m_solveset\u001b[0;34m(f, symbol, domain, _check)\u001b[0m\n\u001b[1;32m    944\u001b[0m                         \u001b[0mresult\u001b[0m \u001b[0;34m+=\u001b[0m \u001b[0m_solve_abs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msymbol\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdomain\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    945\u001b[0m                     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 946\u001b[0;31m                         \u001b[0mresult\u001b[0m \u001b[0;34m+=\u001b[0m \u001b[0m_solve_as_rational\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mequation\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msymbol\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdomain\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    947\u001b[0m                 \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    948\u001b[0m                     \u001b[0mresult\u001b[0m \u001b[0;34m+=\u001b[0m \u001b[0msolver\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mequation\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msymbol\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/solvers/solveset.py\u001b[0m in \u001b[0;36m_solve_as_rational\u001b[0;34m(f, symbol, domain)\u001b[0m\n\u001b[1;32m    488\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mh\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mhas\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msymbol\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    489\u001b[0m         \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 490\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0m_solve_as_poly\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msymbol\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdomain\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    491\u001b[0m         \u001b[0;32mexcept\u001b[0m \u001b[0mNotImplementedError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    492\u001b[0m             \u001b[0;31m# The polynomial formed from g could end up having\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/solvers/solveset.py\u001b[0m in \u001b[0;36m_solve_as_poly\u001b[0;34m(f, symbol, domain)\u001b[0m\n\u001b[1;32m    634\u001b[0m                 \u001b[0mresult\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mConditionSet\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msymbol\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mEq\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[0mdomain\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    635\u001b[0m     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 636\u001b[0;31m         \u001b[0mpoly\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mPoly\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    637\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mpoly\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    638\u001b[0m             \u001b[0mresult\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mConditionSet\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msymbol\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mEq\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[0mdomain\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/polys/polytools.py\u001b[0m in \u001b[0;36m__new__\u001b[0;34m(cls, rep, *gens, **args)\u001b[0m\n\u001b[1;32m    127\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_from_poly\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mrep\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mopt\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    128\u001b[0m             \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 129\u001b[0;31m                 \u001b[0;32mreturn\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_from_expr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mrep\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mopt\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    130\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    131\u001b[0m     \u001b[0;34m@\u001b[0m\u001b[0mclassmethod\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/polys/polytools.py\u001b[0m in \u001b[0;36m_from_expr\u001b[0;34m(cls, rep, opt)\u001b[0m\n\u001b[1;32m    237\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0m_from_expr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mrep\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mopt\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    238\u001b[0m         \u001b[0;34m\"\"\"Construct a polynomial from an expression. \"\"\"\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 239\u001b[0;31m         \u001b[0mrep\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mopt\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_dict_from_expr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mrep\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mopt\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    240\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_from_dict\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mrep\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mopt\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    241\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/polys/polyutils.py\u001b[0m in \u001b[0;36m_dict_from_expr\u001b[0;34m(expr, opt)\u001b[0m\n\u001b[1;32m    344\u001b[0m     \u001b[0;34m\"\"\"Transform an expression into a multinomial form. \"\"\"\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    345\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_commutative\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mFalse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 346\u001b[0;31m         \u001b[0;32mraise\u001b[0m \u001b[0mPolynomialError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'non-commutative expressions are not supported'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    347\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    348\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0m_is_expandable_pow\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mPolynomialError\u001b[0m: non-commutative expressions are not supported"
     ]
    }
   ],
   "source": [
    "sympy.exp(a(x)).series(a(x),n=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 181,
   "metadata": {},
   "outputs": [
    {
     "ename": "KeyboardInterrupt",
     "evalue": "",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m                         Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-181-fecac2a89a49>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0msympy\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mexp\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m+\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mseries\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\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/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36mseries\u001b[0;34m(self, x, x0, n, dir, logx)\u001b[0m\n\u001b[1;32m   2660\u001b[0m             \u001b[0;31m# replace x with an x that has a positive assumption\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2661\u001b[0m             \u001b[0mxpos\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mDummy\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'x'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mpositive\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mTrue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mfinite\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mTrue\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2662\u001b[0;31m             \u001b[0mrv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msubs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mxpos\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mseries\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mxpos\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx0\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mn\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdir\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlogx\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mlogx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2663\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mn\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2664\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0ms\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msubs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mxpos\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ms\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrv\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36mseries\u001b[0;34m(self, x, x0, n, dir, logx)\u001b[0m\n\u001b[1;32m   2667\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2668\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mn\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m  \u001b[0;31m# nseries handling\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2669\u001b[0;31m             \u001b[0ms1\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_eval_nseries\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mn\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlogx\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mlogx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2670\u001b[0m             \u001b[0mo\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0ms1\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mgetO\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0mS\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mZero\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2671\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mo\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/functions/elementary/exponential.py\u001b[0m in \u001b[0;36m_eval_nseries\u001b[0;34m(self, x, n, logx)\u001b[0m\n\u001b[1;32m    422\u001b[0m         \u001b[0mr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mexp\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mexp_series\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msubs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mt\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0marg_series\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0marg0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    423\u001b[0m         \u001b[0mr\u001b[0m \u001b[0;34m+=\u001b[0m \u001b[0mOrder\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mo\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msubs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mt\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0marg_series\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0marg0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\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[0;32m--> 424\u001b[0;31m         \u001b[0mr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mexpand\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    425\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mpowsimp\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdeep\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mTrue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcombine\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m'exp'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    426\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36mexpand\u001b[0;34m(self, deep, modulus, power_base, power_exp, mul, log, multinomial, basic, **hints)\u001b[0m\n\u001b[1;32m   3122\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0muse_hint\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3123\u001b[0m                 \u001b[0mhint\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m'_eval_expand_'\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 3124\u001b[0;31m                 \u001b[0mexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expand_hint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdeep\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mdeep\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   3125\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3126\u001b[0m         \u001b[0;32mwhile\u001b[0m \u001b[0;32mTrue\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   3055\u001b[0m             \u001b[0msargs\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[1;32m   3056\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0marg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 3057\u001b[0;31m                 \u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0marghit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expand_hint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   3058\u001b[0m                 \u001b[0mhit\u001b[0m \u001b[0;34m|=\u001b[0m \u001b[0marghit\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3059\u001b[0m                 \u001b[0msargs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   3055\u001b[0m             \u001b[0msargs\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[1;32m   3056\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0marg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 3057\u001b[0;31m                 \u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0marghit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expand_hint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   3058\u001b[0m                 \u001b[0mhit\u001b[0m \u001b[0;34m|=\u001b[0m \u001b[0marghit\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3059\u001b[0m                 \u001b[0msargs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   3055\u001b[0m             \u001b[0msargs\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[1;32m   3056\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0marg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 3057\u001b[0;31m                 \u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0marghit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expand_hint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   3058\u001b[0m                 \u001b[0mhit\u001b[0m \u001b[0;34m|=\u001b[0m \u001b[0marghit\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3059\u001b[0m                 \u001b[0msargs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   3055\u001b[0m             \u001b[0msargs\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[1;32m   3056\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0marg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 3057\u001b[0;31m                 \u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0marghit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expand_hint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   3058\u001b[0m                 \u001b[0mhit\u001b[0m \u001b[0;34m|=\u001b[0m \u001b[0marghit\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3059\u001b[0m                 \u001b[0msargs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   3063\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3064\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mhasattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 3065\u001b[0;31m             \u001b[0mnewexpr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mgetattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   3066\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mnewexpr\u001b[0m \u001b[0;34m!=\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3067\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mnewexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;32mTrue\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/core/power.py\u001b[0m in \u001b[0;36m_eval_expand_multinomial\u001b[0;34m(self, **hints)\u001b[0m\n\u001b[1;32m   1049\u001b[0m                     \u001b[0;32mreturn\u001b[0m \u001b[0mAdd\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mg\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mf\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mbase\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mbase\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1050\u001b[0m                 \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1051\u001b[0;31m                     \u001b[0mmulti\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mbase\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_eval_expand_multinomial\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   1052\u001b[0m                     \u001b[0;32mif\u001b[0m \u001b[0mmulti\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Add\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1053\u001b[0m                         return Add(*[f*g for f in base.args\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/core/power.py\u001b[0m in \u001b[0;36m_eval_expand_multinomial\u001b[0;34m(self, **hints)\u001b[0m\n\u001b[1;32m   1049\u001b[0m                     \u001b[0;32mreturn\u001b[0m \u001b[0mAdd\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mg\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mf\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mbase\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mbase\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1050\u001b[0m                 \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1051\u001b[0;31m                     \u001b[0mmulti\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mbase\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_eval_expand_multinomial\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   1052\u001b[0m                     \u001b[0;32mif\u001b[0m \u001b[0mmulti\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Add\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1053\u001b[0m                         return Add(*[f*g for f in base.args\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/core/power.py\u001b[0m in \u001b[0;36m_eval_expand_multinomial\u001b[0;34m(self, **hints)\u001b[0m\n\u001b[1;32m   1051\u001b[0m                     \u001b[0mmulti\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mbase\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_eval_expand_multinomial\u001b[0m\u001b[0;34m(\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[0mmulti\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Add\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1053\u001b[0;31m                         return Add(*[f*g for f in base.args\n\u001b[0m\u001b[1;32m   1054\u001b[0m                             for g in multi.args])\n\u001b[1;32m   1055\u001b[0m                     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/core/operations.py\u001b[0m in \u001b[0;36m__new__\u001b[0;34m(cls, *args, **options)\u001b[0m\n\u001b[1;32m     43\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     44\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 45\u001b[0;31m         \u001b[0mc_part\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnc_part\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0morder_symbols\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mflatten\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     46\u001b[0m         \u001b[0mis_commutative\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mnc_part\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     47\u001b[0m         \u001b[0mobj\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_from_args\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mc_part\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mnc_part\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mis_commutative\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/core/add.py\u001b[0m in \u001b[0;36mflatten\u001b[0;34m(cls, seq)\u001b[0m\n\u001b[1;32m    252\u001b[0m                 \u001b[0;32mfor\u001b[0m \u001b[0mo\u001b[0m \u001b[0;32min\u001b[0m \u001b[0morder_factors\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    253\u001b[0m                     \u001b[0;31m# x + O(x) -> O(x)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 254\u001b[0;31m                     \u001b[0;32mif\u001b[0m \u001b[0mo\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcontains\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mt\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    255\u001b[0m                         \u001b[0mt\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    256\u001b[0m                         \u001b[0;32mbreak\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/series/order.py\u001b[0m in \u001b[0;36mcontains\u001b[0;34m(self, expr)\u001b[0m\n\u001b[1;32m    407\u001b[0m                         \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mr\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    408\u001b[0m                             \u001b[0;32mreturn\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mr\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 409\u001b[0;31m         \u001b[0mobj\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1\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    410\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcontains\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mobj\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    411\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/series/order.py\u001b[0m in \u001b[0;36m__new__\u001b[0;34m(cls, expr, *args, **kwargs)\u001b[0m\n\u001b[1;32m    215\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    216\u001b[0m             \u001b[0;32melif\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 217\u001b[0;31m                 \u001b[0mexpr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_leading_term\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    218\u001b[0m                 \u001b[0mexpr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_independent\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mas_Add\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mFalse\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    219\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36mas_leading_term\u001b[0;34m(self, *symbols)\u001b[0m\n\u001b[1;32m   2943\u001b[0m         \u001b[0mobj\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_eval_as_leading_term\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[1;32m   2944\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mobj\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2945\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0mpowsimp\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mobj\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdeep\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mTrue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcombine\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m'exp'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2946\u001b[0m         \u001b[0;32mraise\u001b[0m \u001b[0mNotImplementedError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'as_leading_term(%s, %s)'\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2947\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/simplify/powsimp.py\u001b[0m in \u001b[0;36mpowsimp\u001b[0;34m(expr, deep, combine, force, measure)\u001b[0m\n\u001b[1;32m    116\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    117\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mdeep\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Add\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Mul\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0m_y\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 118\u001b[0;31m         \u001b[0mexpr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mrecurse\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mw\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mw\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\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    119\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    120\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Pow\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/simplify/powsimp.py\u001b[0m in \u001b[0;36m<listcomp>\u001b[0;34m(.0)\u001b[0m\n\u001b[1;32m    116\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    117\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mdeep\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Add\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Mul\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0m_y\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 118\u001b[0;31m         \u001b[0mexpr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mrecurse\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mw\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mw\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\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    119\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    120\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Pow\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/simplify/powsimp.py\u001b[0m in \u001b[0;36mrecurse\u001b[0;34m(arg, **kwargs)\u001b[0m\n\u001b[1;32m    107\u001b[0m         \u001b[0m_force\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mkwargs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'force'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mforce\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    108\u001b[0m         \u001b[0m_measure\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mkwargs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'measure'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mmeasure\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 109\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mpowsimp\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0m_deep\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0m_combine\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0m_force\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0m_measure\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    110\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    111\u001b[0m     \u001b[0mexpr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msympify\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/simplify/powsimp.py\u001b[0m in \u001b[0;36mpowsimp\u001b[0;34m(expr, deep, combine, force, measure)\u001b[0m\n\u001b[1;32m    116\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    117\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mdeep\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Add\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Mul\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0m_y\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 118\u001b[0;31m         \u001b[0mexpr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mrecurse\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mw\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mw\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\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    119\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    120\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Pow\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/simplify/powsimp.py\u001b[0m in \u001b[0;36m<listcomp>\u001b[0;34m(.0)\u001b[0m\n\u001b[1;32m    116\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    117\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mdeep\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Add\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Mul\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0m_y\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 118\u001b[0;31m         \u001b[0mexpr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mrecurse\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mw\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mw\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\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    119\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    120\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Pow\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/simplify/powsimp.py\u001b[0m in \u001b[0;36mrecurse\u001b[0;34m(arg, **kwargs)\u001b[0m\n\u001b[1;32m    107\u001b[0m         \u001b[0m_force\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mkwargs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'force'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mforce\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    108\u001b[0m         \u001b[0m_measure\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mkwargs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'measure'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mmeasure\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 109\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mpowsimp\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0m_deep\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0m_combine\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0m_force\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0m_measure\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    110\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    111\u001b[0m     \u001b[0mexpr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msympify\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/simplify/powsimp.py\u001b[0m in \u001b[0;36mpowsimp\u001b[0;34m(expr, deep, combine, force, measure)\u001b[0m\n\u001b[1;32m    116\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    117\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mdeep\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Add\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Mul\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0m_y\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 118\u001b[0;31m         \u001b[0mexpr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mrecurse\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mw\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mw\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\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    119\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    120\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Pow\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/core/function.py\u001b[0m in \u001b[0;36m__new__\u001b[0;34m(cls, expr, *variables, **kwargs)\u001b[0m\n\u001b[1;32m   1105\u001b[0m         \u001b[0;32mfrom\u001b[0m \u001b[0msympy\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mInteger\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1106\u001b[0m         \u001b[0;32mfrom\u001b[0m \u001b[0msympy\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mtensor\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0marray\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mArray\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mNDimArray\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mderive_by_array\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1107\u001b[0;31m         \u001b[0;32mfrom\u001b[0m \u001b[0msympy\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mutilities\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmisc\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mfilldedent\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1108\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1109\u001b[0m         \u001b[0mexpr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msympify\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/importlib/_bootstrap.py\u001b[0m in \u001b[0;36m_handle_fromlist\u001b[0;34m(module, fromlist, import_)\u001b[0m\n",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m: "
     ]
    }
   ],
   "source": [
    "sympy.exp(a(x)+b(x)).series(x=x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$a{\\left (z \\right )} b{\\left (z \\right )} - b{\\left (z \\right )} a{\\left (z \\right )}$$"
      ],
      "text/plain": [
       "a(z)⋅b(z) - b(z)⋅a(z)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import *\n",
    "var('z')\n",
    "a = Function('a', commutative=False)\n",
    "b = Function('b', commutative=False)\n",
    "a(z)*b(z)-b(z)*a(z)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'1.2'"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.__version__"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [],
   "source": [
    "class PoissonOperator(Function):\n",
    "    @classmethod\n",
    "    def eval(cls, f,g,x,p):\n",
    "        if a == 0:\n",
    "            return Integer(0)\n",
    "        else:\n",
    "            summ = 0\n",
    "            for i in range(len(x)):\n",
    "                summ += f.diff(x[i]) * g.diff(p[i]) \n",
    "                summ -= f.diff(p[i]) * g.diff(x[i])\n",
    "\n",
    "            return summ\n",
    "            "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 177,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- \\frac{\\partial}{\\partial p} H{\\left (x,p \\right )} \\frac{\\partial}{\\partial x} g{\\left (x,p \\right )} + \\frac{\\partial}{\\partial x} H{\\left (x,p \\right )} \\frac{\\partial}{\\partial p} g{\\left (x,p \\right )}$$"
      ],
      "text/plain": [
       "  ∂           ∂             ∂           ∂          \n",
       "- ──(H(x, p))⋅──(g(x, p)) + ──(H(x, p))⋅──(g(x, p))\n",
       "  ∂p          ∂x            ∂x          ∂p         "
      ]
     },
     "execution_count": 177,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "PoissonOperator(H,g,[x],[p])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$$- \\frac{\\partial}{\\partial p} H{\\left (x,p \\right )} \\left(- \\frac{\\partial}{\\partial p} H{\\left (x,p \\right )} \\frac{\\partial^{2}}{\\partial x^{2}} g{\\left (x,p \\right )} + \\frac{\\partial}{\\partial x} H{\\left (x,p \\right )} \\frac{\\partial^{2}}{\\partial x\\partial p} g{\\left (x,p \\right )} + \\frac{\\partial^{2}}{\\partial x^{2}} H{\\left (x,p \\right )} \\frac{\\partial}{\\partial p} g{\\left (x,p \\right )} - \\frac{\\partial^{2}}{\\partial x\\partial p} H{\\left (x,p \\right )} \\frac{\\partial}{\\partial x} g{\\left (x,p \\right )}\\right) + \\frac{\\partial}{\\partial x} H{\\left (x,p \\right )} \\left(- \\frac{\\partial}{\\partial p} H{\\left (x,p \\right )} \\frac{\\partial^{2}}{\\partial x\\partial p} g{\\left (x,p \\right )} - \\frac{\\partial^{2}}{\\partial p^{2}} H{\\left (x,p \\right )} \\frac{\\partial}{\\partial x} g{\\left (x,p \\right )} + \\frac{\\partial}{\\partial x} H{\\left (x,p \\right )} \\frac{\\partial^{2}}{\\partial p^{2}} g{\\left (x,p \\right )} + \\frac{\\partial^{2}}{\\partial x\\partial p} H{\\left (x,p \\right )} \\frac{\\partial}{\\partial p} g{\\left (x,p \\right )}\\right)$$"
      ],
      "text/plain": [
       "              ⎛                2                           2               2  \n",
       "  ∂           ⎜  ∂            ∂             ∂             ∂               ∂   \n",
       "- ──(H(x, p))⋅⎜- ──(H(x, p))⋅───(g(x, p)) + ──(H(x, p))⋅─────(g(x, p)) + ───(H\n",
       "  ∂p          ⎜  ∂p            2            ∂x          ∂x ∂p              2  \n",
       "              ⎝              ∂x                                          ∂x   \n",
       "\n",
       "                         2                      ⎞               ⎛             \n",
       "        ∂               ∂            ∂          ⎟   ∂           ⎜  ∂          \n",
       "(x, p))⋅──(g(x, p)) - ─────(H(x, p))⋅──(g(x, p))⎟ + ──(H(x, p))⋅⎜- ──(H(x, p))\n",
       "        ∂p            ∂x ∂p          ∂x         ⎟   ∂x          ⎜  ∂p         \n",
       "                                                ⎠               ⎝             \n",
       "\n",
       "    2               2                                      2               2  \n",
       "   ∂               ∂           ∂             ∂            ∂               ∂   \n",
       "⋅─────(g(x, p)) - ───(H(x, p))⋅──(g(x, p)) + ──(H(x, p))⋅───(g(x, p)) + ─────(\n",
       " ∂x ∂p              2          ∂x            ∂x            2            ∂x ∂p \n",
       "                  ∂p                                     ∂p                   \n",
       "\n",
       "                    ⎞\n",
       "         ∂          ⎟\n",
       "H(x, p))⋅──(g(x, p))⎟\n",
       "         ∂p         ⎟\n",
       "                    ⎠"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "PoissonOperator(H,PoissonOperator(H,g,[x],[p]),[x],[p])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x\n",
      "p\n"
     ]
    }
   ],
   "source": [
    "for i in H.args:\n",
    "    print(i)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "H"
      ]
     },
     "execution_count": 102,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H.func"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "class PoissonOperator():\n",
    "    def __new__(cls, arg):\n",
    "        coords = arg.args[:len(arg.args)/2]\n",
    "        momenta = arg.args[len(arg.args)/2:]\n",
    "#         if hasattr(arg, 'coordinates'):\n",
    "#             c = arg.coordinates\n",
    "#         elif hasattr(arg, 'conjugate') and hasattr(arg, 'transpose'):\n",
    "#             p = arg.conjugate().transpose()\n",
    "        if coords is not None:\n",
    "            summ = 0\n",
    "            for i in range(len(coords)):\n",
    "                summ += arg.func.diff(coords[i]) * g.diff(p[i]) \n",
    "                summ -= arg.func.diff(p[i]) * g.diff(x[i])\n",
    "            return summ\n",
    "        return Expr.__new__(cls, arg)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {},
   "outputs": [],
   "source": [
    "class Operator(Function):\n",
    "    @classmethod\n",
    "    def eval(cls, f,x):\n",
    "        if a == 0:\n",
    "            return Integer(0)\n",
    "        else:\n",
    "            summ = 0\n",
    "            for i in range(len(x)):\n",
    "                summ += f.diff(x[i]) * g.diff(p[i]) \n",
    "                summ -= f.diff(p[i]) * g.diff(x[i])\n",
    "\n",
    "            return summ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "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-99-b4ff02d1698e>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mOperator\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mH\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/core/function.py\u001b[0m in \u001b[0;36m__new__\u001b[0;34m(cls, *args, **options)\u001b[0m\n\u001b[1;32m    439\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    440\u001b[0m         \u001b[0mevaluate\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0moptions\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'evaluate'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mglobal_evaluate\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[0m\n\u001b[0;32m--> 441\u001b[0;31m         \u001b[0mresult\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msuper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mFunction\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__new__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0moptions\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    442\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mevaluate\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mresult\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0mresult\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    443\u001b[0m             \u001b[0mpr2\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mmin\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_should_evalf\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mresult\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/tommertens/anaconda2/envs/py35/lib/python3.5/site-packages/sympy/core/function.py\u001b[0m in \u001b[0;36m__new__\u001b[0;34m(cls, *args, **options)\u001b[0m\n\u001b[1;32m    248\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    249\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mevaluate\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 250\u001b[0;31m             \u001b[0mevaluated\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0meval\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    251\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mevaluated\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    252\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0mevaluated\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m<ipython-input-98-4a1d632b1868>\u001b[0m in \u001b[0;36meval\u001b[0;34m(cls, f, x)\u001b[0m\n\u001b[1;32m      7\u001b[0m             \u001b[0msumm\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;36m0\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      8\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mlen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\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[0;32m----> 9\u001b[0;31m                 \u001b[0msumm\u001b[0m \u001b[0;34m+=\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdiff\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdiff\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\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     10\u001b[0m                 \u001b[0msumm\u001b[0m \u001b[0;34m-=\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdiff\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdiff\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     11\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mTypeError\u001b[0m: 'Symbol' object does not support indexing"
     ]
    }
   ],
   "source": [
    "Operator(H,[x])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Baker-Campbell-Hausdorff\n",
    "\n",
    "$$F_{ij}= \\frac{1}{(j-i)!} = \\exp\\left(\\delta_{i+1,j}\\right)$$\n",
    "\n",
    "$$G_{ij}= \\frac{1}{(j-i)!}\\prod_{k=i}^{j-1}\\sigma_k = \\exp\\left(\\delta_{i+1,j} \\sigma_i\\right)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f + g\n",
      "f*g/2 - g*f/2\n",
      "f*f*g/12 - f*g*f/6 + f*g*g/12 + g*f*f/12 - g*f*g/6 + g*g*f/12\n",
      "f*f*g*g/24 - f*g*f*g/12 + g*f*g*f/12 - g*g*f*f/24\n"
     ]
    }
   ],
   "source": [
    "n = 4\n",
    "def p(n):\n",
    "    F = sympy.Matrix(np.zeros((n+1,n+1)))\n",
    "    for i in range(n+1):\n",
    "        for j in range(n+1):\n",
    "            if (j-i)>=0:\n",
    "    #             F[i,j] = 1/(np.math.factorial(j-i))\n",
    "                F[i,j] = sympy.Rational(1,int(fac(j-i))) # keep it rational !!!!\n",
    "\n",
    "\n",
    "    \n",
    "    G = sympy.Matrix(np.zeros((n+1,n+1)))\n",
    "    for i in range(n+1):\n",
    "        for j in range(n+1):\n",
    "            if (j-i)>=0:\n",
    "                if i==j:\n",
    "                    sigprod = 1\n",
    "                else:\n",
    "                    l = sympy.symbols('s{}:{}'.format(i,j))\n",
    "                    sigprod = reduce(sympy.Mul,l)\n",
    "                G[i,j] = sympy.Mul(sympy.Rational(1,int(np.math.factorial(j-i))),sigprod, evaluate=False)\n",
    "\n",
    "\n",
    "\n",
    "    qthpower = sympy.Matrix(np.eye(n+1))\n",
    "\n",
    "    FGm1 = sympy.nsimplify(F*G - qthpower)\n",
    "\n",
    "    FGm1\n",
    "\n",
    "\n",
    "    summ = 0\n",
    "\n",
    "    for q in range(1,n+1):\n",
    "        qthpower = sympy.nsimplify(sympy.MatMul(qthpower, FGm1, evaluate=False))\n",
    "\n",
    "        summ += sympy.nsimplify((-1)**(q-1)/q * qthpower)[0,n] # q-1 because mathematica starts at 1 and not 0, there for sign change\n",
    "\n",
    "    return sympy.expand(summ)\n",
    "\n",
    "def translated(n):\n",
    "    l = sympy.symbols('s{}:{}'.format(0,n))\n",
    "    temp = reduce(functools.partial(sympy.Mul,evaluate=False),[k*k for k in l])\n",
    "    temp = sympy.expand(sympy.Mul(temp, p(n), evaluate=False),evaluate=False)\n",
    "    with sympy.evaluate(False):\n",
    "        ln = sympy.symbols('r{}:{}'.format(0,n), commutative=False)\n",
    "        for i, s in enumerate(l):\n",
    "            temp = temp.subs(l[i],ln[i])\n",
    "        for i in range(n):\n",
    "            g = sympy.Symbol('g',commutative=False)\n",
    "            f = sympy.Symbol('f',commutative=False)\n",
    "            temp = temp.subs(ln[i]**3,g)\n",
    "            temp = temp.subs(ln[i]**2,f)\n",
    "    \n",
    "    return temp\n",
    "   \n",
    "\n",
    "print(translated(1))\n",
    "print(translated(2))\n",
    "print(translated(3))\n",
    "print(translated(4))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAC4AAAASBAMAAADWL/HSAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJl2IquJVETdZu8yu83OyatpAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAwklEQVQYGWNgYGAQMnIAkpggrL4AU5CBgfGHBESY3QBFmv0DlIsmzjYBqzhX9FQFiARcfZFxLVCEH2YsTJztAssnoLj/AYhyBph4vQDDd6BQPViY7d27l/vevQsAcvYy8HwBUplgcSABVc/zi4EZ5JSraOJcXxh4FYBim9HEgf7hLwB5F02c8QtDvAMDA8dHNHGGVjOQlWwX0MUZGC4xMJTyLYCJMzqAWTxgo+dKwoRhdL4AL9AXi1VgfBhdulidgQEA93soR2MLku0AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$f + g$$"
      ],
      "text/plain": [
       "f + g"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "translated(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( - \\frac{1}{6}, \\quad f, \\quad g, \\quad f\\right )$$"
      ],
      "text/plain": [
       "(-1/6, f, g, f)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "translated(3).args[0].args"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f*f*g/12 - f*g*f/6 + f*g*g/12 + g*f*f/12 - g*f*g/6 + g*g*f/12\n",
      "-f*g*f/6\n",
      "-1/6\n",
      "f\n",
      "g\n",
      "f\n",
      "-g*f*g/6\n",
      "-1/6\n",
      "g\n",
      "f\n",
      "g\n",
      "f*f*g/12\n",
      "1/12\n",
      "f\n",
      "f\n",
      "g\n",
      "f*g*g/12\n",
      "1/12\n",
      "f\n",
      "g\n",
      "g\n",
      "g*f*f/12\n",
      "1/12\n",
      "g\n",
      "f\n",
      "f\n",
      "g*g*f/12\n",
      "1/12\n",
      "g\n",
      "g\n",
      "f\n"
     ]
    }
   ],
   "source": [
    "for a in sympy.preorder_traversal(translated(3)):\n",
    "    print(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'f' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-12-e263ab58098a>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mtranslated\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m4\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msubs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\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;31mNameError\u001b[0m: name 'f' is not defined"
     ]
    }
   ],
   "source": [
    "translated(4).subs(f,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( f, \\quad g, \\quad f\\right )$$"
      ],
      "text/plain": [
       "(f, g, f)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "translated(4).args[0].args[1:-1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "la = translated(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-f*f*g/6 - f*g*g/6 + f*f*g/12 + f*g*g/12 + f*f*g/12 + f*g*g/12]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "la.as_ordered_factors()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "a0*a1*a2\n"
     ]
    }
   ],
   "source": [
    "print(reduce(sympy.Mul,sympy.symbols('a{}:{}'.format(0,3))))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "p = sympy.symbols('p')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "PoissionBracket(x,x,x,p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# symbols for drift\n",
    "delta, px, py, L = sympy.symbols('delta px py, L')\n",
    "\n",
    "# drift Lie operator\n",
    "f_drift = - L * sympy.sqrt((1+delta**2 - px**2 - py**2))\n",
    "exp_f_drift = sympy.exp(f_drift)\n",
    "\n",
    "def exp_f_drift_order(n):\n",
    "    return exp_f_drift.expand()\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Algebra classes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "class LieAlgElem:\n",
    "    \n",
    "    def __init__(self,expr):\n",
    "        self.expr = expr\n",
    "        \n",
    "    @staticmethod\n",
    "    def LieBracket(f,g,x,p):\n",
    "        if not isinstance(x, list):\n",
    "            x = [x]\n",
    "        if not isinstance(p, list):\n",
    "            p = [p]\n",
    "\n",
    "        assert(len(x)==len(p))\n",
    "\n",
    "        summ = 0\n",
    "        for i in range(len(x)):\n",
    "            summ +=  f.diff(x[i]) * g.diff(p[i]) - f.diff(p[i]) * g.diff(x[i])\n",
    "\n",
    "        return summ\n",
    "    \n",
    "    def __add__(self, other):\n",
    "        return sympy.Add(self,other, evaluate=False)\n",
    "    \n",
    "    def __mul__(self, other, order_bracket, order_expansion):\n",
    "        pass\n",
    "        \n",
    "        "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPBAMAAAArJJMAAAAAHlBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACGjDitAAAACXRSTlMAVO8Qq5l2zWYZcMvdAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAHUlEQVQIHWNgAANGZQYGk5DJQDYbqQSr03QPsBkAJYgIYEZbtZEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1$$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = sympy.Symbol('x')\n",
    "f = sympy.Function('f')(x)\n",
    "f.diff(x).subs({f:x}).doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "x,p = sympy.symbols('x p')\n",
    "f = sympy.Function('f')(x,p)\n",
    "g = sympy.Function('g')(x,p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABQAAAAOBAMAAADd6iHDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12VGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAf0lEQVQIHWNgVDJ2YICAMAb2H1BmKgPDTChzFgNDvgOEvT8AzgQKrA9gPZPYUwNkcXxnCGd4dWA1kMmVwFDKUB9wEchUYmAIYNgMZDDwKIDIPyDiEgOjAAPLFyCLRYGBR+AqzweGowxS5UVZHH+ZPzA+YNj///8v1pyjh94xAABbhSE1xl33oAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$2 x$$"
      ],
      "text/plain": [
       "2⋅x"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.diff(x).subs({f:x**2}).doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 116,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x.diff(p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{\\partial}{\\partial p} g{\\left (x,p \\right )}$$"
      ],
      "text/plain": [
       "∂          \n",
       "──(g(x, p))\n",
       "∂p         "
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.simplify(LieBracket(f,g,x,p).subs(f,x).doit())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Test"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Drift"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-delta/sqrt(delta**2 - px**2 - py**2 + 1)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H_drift.diff(delta)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-L*px/sqrt(delta**2 - px**2 - py**2 + 1)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x,y = sympy.symbols('x y')\n",
    "\n",
    "PoissionBracket(f_drift,x,[x,y],[px,py])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "exp(-L*sqrt(delta**2 - px**2 - py**2 + 1))"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp_f_drift_order(0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-15*L/(16*sqrt(-py**2 + 1)) + 5*L*sqrt(-py**2 + 1)/(16*(py**4 - 2*py**2 + 1)) - L*sqrt(-py**2 + 1)/(16*(-py**6 + 3*py**4 - 3*py**2 + 1)) - 5*L*sqrt(-py**2 + 1)/16 + 15*L*py**2/(16*sqrt(-py**2 + 1)) - 5*L*py**2*sqrt(-py**2 + 1)/(8*(py**4 - 2*py**2 + 1)) + 3*L*py**2*sqrt(-py**2 + 1)/(16*(-py**6 + 3*py**4 - 3*py**2 + 1)) + 5*L*py**4*sqrt(-py**2 + 1)/(16*(py**4 - 2*py**2 + 1)) - 3*L*py**4*sqrt(-py**2 + 1)/(16*(-py**6 + 3*py**4 - 3*py**2 + 1)) + L*py**6*sqrt(-py**2 + 1)/(16*(-py**6 + 3*py**4 - 3*py**2 + 1)) + O(px)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.series(f_drift.subs(delta,0),px,n=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "L*px + O(px**2)"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "PoissionBracket(sympy.series(-L*sympy.sqrt(1+px**2),px,n=3), x,[x,y],[px,py])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 216,
   "metadata": {},
   "outputs": [],
   "source": [
    "def gen(q0,p0,dt =1e-3,steps=100,K=.5):\n",
    "    itsteps = iter(steps)\n",
    "    Q=[q0]\n",
    "    P=[p0]\n",
    "    while True:\n",
    "        try:\n",
    "            q1 = (q0 + K* np.sin(p0)) % (2 * np.pi)\n",
    "            p1 = (p0 + q1)% (2 * np.pi)\n",
    "            Q.append(q1)\n",
    "            P.append(p1)\n",
    "            q0 = q1\n",
    "            p0 = p1\n",
    "        except StopIteration:\n",
    "            break\n",
    "    return list(zip(Q,P))\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 217,
   "metadata": {},
   "outputs": [],
   "source": [
    "qs = np.linspace(0,2*np.pi,10)\n",
    "ps = np.linspace(0,2*np.pi,10)\n",
    "\n",
    "grid = np.meshgrid(qs,ps)\n",
    "coords= "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 218,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[array([[0.        , 0.6981317 , 1.3962634 , 2.0943951 , 2.7925268 ,\n",
       "         3.4906585 , 4.1887902 , 4.88692191, 5.58505361, 6.28318531],\n",
       "        [0.        , 0.6981317 , 1.3962634 , 2.0943951 , 2.7925268 ,\n",
       "         3.4906585 , 4.1887902 , 4.88692191, 5.58505361, 6.28318531],\n",
       "        [0.        , 0.6981317 , 1.3962634 , 2.0943951 , 2.7925268 ,\n",
       "         3.4906585 , 4.1887902 , 4.88692191, 5.58505361, 6.28318531],\n",
       "        [0.        , 0.6981317 , 1.3962634 , 2.0943951 , 2.7925268 ,\n",
       "         3.4906585 , 4.1887902 , 4.88692191, 5.58505361, 6.28318531],\n",
       "        [0.        , 0.6981317 , 1.3962634 , 2.0943951 , 2.7925268 ,\n",
       "         3.4906585 , 4.1887902 , 4.88692191, 5.58505361, 6.28318531],\n",
       "        [0.        , 0.6981317 , 1.3962634 , 2.0943951 , 2.7925268 ,\n",
       "         3.4906585 , 4.1887902 , 4.88692191, 5.58505361, 6.28318531],\n",
       "        [0.        , 0.6981317 , 1.3962634 , 2.0943951 , 2.7925268 ,\n",
       "         3.4906585 , 4.1887902 , 4.88692191, 5.58505361, 6.28318531],\n",
       "        [0.        , 0.6981317 , 1.3962634 , 2.0943951 , 2.7925268 ,\n",
       "         3.4906585 , 4.1887902 , 4.88692191, 5.58505361, 6.28318531],\n",
       "        [0.        , 0.6981317 , 1.3962634 , 2.0943951 , 2.7925268 ,\n",
       "         3.4906585 , 4.1887902 , 4.88692191, 5.58505361, 6.28318531],\n",
       "        [0.        , 0.6981317 , 1.3962634 , 2.0943951 , 2.7925268 ,\n",
       "         3.4906585 , 4.1887902 , 4.88692191, 5.58505361, 6.28318531]]),\n",
       " array([[0.        , 0.        , 0.        , 0.        , 0.        ,\n",
       "         0.        , 0.        , 0.        , 0.        , 0.        ],\n",
       "        [0.6981317 , 0.6981317 , 0.6981317 , 0.6981317 , 0.6981317 ,\n",
       "         0.6981317 , 0.6981317 , 0.6981317 , 0.6981317 , 0.6981317 ],\n",
       "        [1.3962634 , 1.3962634 , 1.3962634 , 1.3962634 , 1.3962634 ,\n",
       "         1.3962634 , 1.3962634 , 1.3962634 , 1.3962634 , 1.3962634 ],\n",
       "        [2.0943951 , 2.0943951 , 2.0943951 , 2.0943951 , 2.0943951 ,\n",
       "         2.0943951 , 2.0943951 , 2.0943951 , 2.0943951 , 2.0943951 ],\n",
       "        [2.7925268 , 2.7925268 , 2.7925268 , 2.7925268 , 2.7925268 ,\n",
       "         2.7925268 , 2.7925268 , 2.7925268 , 2.7925268 , 2.7925268 ],\n",
       "        [3.4906585 , 3.4906585 , 3.4906585 , 3.4906585 , 3.4906585 ,\n",
       "         3.4906585 , 3.4906585 , 3.4906585 , 3.4906585 , 3.4906585 ],\n",
       "        [4.1887902 , 4.1887902 , 4.1887902 , 4.1887902 , 4.1887902 ,\n",
       "         4.1887902 , 4.1887902 , 4.1887902 , 4.1887902 , 4.1887902 ],\n",
       "        [4.88692191, 4.88692191, 4.88692191, 4.88692191, 4.88692191,\n",
       "         4.88692191, 4.88692191, 4.88692191, 4.88692191, 4.88692191],\n",
       "        [5.58505361, 5.58505361, 5.58505361, 5.58505361, 5.58505361,\n",
       "         5.58505361, 5.58505361, 5.58505361, 5.58505361, 5.58505361],\n",
       "        [6.28318531, 6.28318531, 6.28318531, 6.28318531, 6.28318531,\n",
       "         6.28318531, 6.28318531, 6.28318531, 6.28318531, 6.28318531]])]"
      ]
     },
     "execution_count": 218,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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