jankoslavic/stack-34242675

## stack-34242675
{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "p, r, c, p, y, lam, f = symbols('p r c p y lambda f')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEYAAAAwBAMAAABeexFrAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mSJE3e9UMqtm\nzbsXyEShAAABBElEQVQ4EWNgwAYYX2MTRROTT0ATwMLl3oBFEE2I7SOaADZuMTZBNDF1ND4WLuv+\nBVhEIUKc74DgIQPDWj6CjuYqxOFoRmWTAKgFGQsYarDalR7AqgCVeMbAcBybGtZOBpEF2CSQxLh7\njA8icbEy+Qn6hIGBXwGrVhRBDqAaAxQRTA7zBQYRmN8xZaEiJjYHcMqNSlAaAv8JA0qtIFI/tQsQ\nXgE8FkPyIut7fGoYQAUI45P9eNVAChB7fGqgBQhWNWgFCIYapMIBVoBgqEEqHGAFCLoa5MIBVoBA\n1MiWg8AtoE+xFQ7o5mArHDDUKGCGO7oabIUDuhoshYN3f/UBVMPJKRwA87lduhFOJ+kAAAAASUVO\nRK5CYII=\n",
      "text/latex": [
       "$$\\frac{e^{- \\lambda}}{1 - e^{- \\lambda}}$$"
      ],
      "text/plain": [
       "   -λ  \n",
       "  ℯ    \n",
       "───────\n",
       "     -λ\n",
       "1 - ℯ  "
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "constant1 = exp(-lam) / (1 - exp(-lam))\n",
    "constant1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKYAAAAxBAMAAACixc+/AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nzP4CGwH3xT6FJz8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{\\left(- \\lambda + e^{\\lambda} - 1\\right) e^{- \\lambda}}{1 - e^{- \\lambda}}$$"
      ],
      "text/plain": [
       "⎛      λ    ⎞  -λ\n",
       "⎝-λ + ℯ  - 1⎠⋅ℯ  \n",
       "─────────────────\n",
       "          -λ     \n",
       "     1 - ℯ       "
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "constant2 = constant1 * (exp(lam) -  1 - lam)\n",
    "constant2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAkQAAAAxBAMAAADJvqsyAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$1 - \\frac{\\left(- \\lambda + e^{\\lambda} - 1\\right) e^{- \\lambda}}{1 - e^{- \\lambda}} + \\frac{e^{- \\lambda}}{1 - e^{- \\lambda}} \\left(- \\lambda \\left(- f + 1\\right) + e^{\\lambda \\left(- f + 1\\right)} - 1\\right)$$"
      ],
      "text/plain": [
       "    ⎛      λ    ⎞  -λ   ⎛               λ⋅(-f + 1)    ⎞  -λ\n",
       "    ⎝-λ + ℯ  - 1⎠⋅ℯ     ⎝-λ⋅(-f + 1) + ℯ           - 1⎠⋅ℯ  \n",
       "1 - ───────────────── + ───────────────────────────────────\n",
       "              -λ                           -λ              \n",
       "         1 - ℯ                        1 - ℯ                "
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eta = 1 - constant2 + constant1 * (exp(lam * (1-f)) - 1 - lam * (1 - f))\n",
    "eta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADkAAAAsBAMAAAAtLQ2eAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKuJ70RUuyJ23TLN\nmWb8I+xeAAABDElEQVQ4EWNgQAKMSkgcTKa8AaYYQoTJAcHGZLF+wxRDEnFHYmMy7QUwxRAi/RMQ\nbEyW/AJMMbgIY+VvOBuT0dPwF1MQJsKxmuEKjI1Jbwlg2J+AKQwR4f7HwMBVgEuWHegdvh+4ZEfF\nGf7jAR+oED7iCvgMYfyKT5YBfw6KF0DXfBBJgAsY8yig/SMSl3kBEgfIFL+ELNuEnpA5kWS5DxSh\n6mWAyjJazm1g2CWA7iyobFQDhwGrNwPXA1TNEFkONwbZDewBDHwLsMky+c18CBeXWQUCp0F8iF5+\nBbgcCgMqa4AiCOdAZFmAsujhBFICkeU8wCDbANeBYEBkGebOe4AQg7N4VX7pwTk4GAAyglHXZoJP\n2QAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{\\lambda}{e^{\\lambda} - 1}$$"
      ],
      "text/plain": [
       "  λ   \n",
       "──────\n",
       " λ    \n",
       "ℯ  - 1"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "etaOfR = limit(eta, f,  1)\n",
    "etaOfR"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/latex": [
       "$$\\frac{y}{p} \\left(- c + p\\right) \\left(1 - \\frac{\\left(- \\lambda + e^{\\lambda} - 1\\right) e^{- \\lambda}}{1 - e^{- \\lambda}} + \\frac{e^{- \\lambda}}{1 - e^{- \\lambda}} \\left(- \\lambda \\left(- f + 1\\right) + e^{\\lambda \\left(- f + 1\\right)} - 1\\right)\\right) = \\frac{\\lambda y \\left(- c + r\\right)}{r \\left(e^{\\lambda} - 1\\right)}$$"
      ],
      "text/plain": [
       "           ⎛    ⎛      λ    ⎞  -λ   ⎛               λ⋅(-f + 1)    ⎞  -λ⎞      \n",
       "           ⎜    ⎝-λ + ℯ  - 1⎠⋅ℯ     ⎝-λ⋅(-f + 1) + ℯ           - 1⎠⋅ℯ  ⎟      \n",
       "y⋅(-c + p)⋅⎜1 - ───────────────── + ───────────────────────────────────⎟      \n",
       "           ⎜              -λ                           -λ              ⎟      \n",
       "           ⎝         1 - ℯ                        1 - ℯ                ⎠   λ⋅y\n",
       "──────────────────────────────────────────────────────────────────────── = ───\n",
       "                                   p                                          \n",
       "                                                                            r⋅\n",
       "\n",
       "         \n",
       "         \n",
       "         \n",
       "         \n",
       "⋅(-c + r)\n",
       "─────────\n",
       "⎛ λ    ⎞ \n",
       "⎝ℯ  - 1⎠ "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expression1 = Eq(y / p * (p - c) * eta, \n",
    "                 y / r * (r - c) * etaOfR)\n",
    "expression1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA0cAAAAyCAMAAACgY1BeAAAAM1BMVEX///8AAAAAAAAAAAAAAAAA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\na68Q33VfAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left [ \\frac{1}{\\lambda r \\left(c - p\\right)} \\left(c \\lambda p + c r - \\lambda p r - p r + r \\left(c - p\\right) \\operatorname{LambertW}{\\left (- e^{\\frac{1}{r \\left(c - p\\right)} \\left(- c \\lambda p - c r + \\lambda p r + \\lambda r \\left(c - p\\right) + p r\\right)} \\right )}\\right)\\right ]$$"
      ],
      "text/plain": [
       "⎡                                              ⎛  -c⋅λ⋅p - c⋅r + λ⋅p⋅r + λ⋅r⋅(\n",
       "⎢                                              ⎜  ────────────────────────────\n",
       "⎢                                              ⎜                 r⋅(c - p)    \n",
       "⎢c⋅λ⋅p + c⋅r - λ⋅p⋅r - p⋅r + r⋅(c - p)⋅LambertW⎝-ℯ                            \n",
       "⎢─────────────────────────────────────────────────────────────────────────────\n",
       "⎣                                       λ⋅r⋅(c - p)                           \n",
       "\n",
       "c - p) + p⋅r⎞⎤\n",
       "────────────⎟⎥\n",
       "            ⎟⎥\n",
       "            ⎠⎥\n",
       "─────────────⎥\n",
       "             ⎦"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol = solve(expression1, f)\n",
    "sol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAzYAAAAvCAMAAAAGoK+9AAAAM1BMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\frac{1}{\\lambda r \\left(c - p\\right)} \\left(c \\lambda p + c r - \\lambda p r - p r + r \\left(c - p\\right) \\operatorname{LambertW}{\\left (- e^{\\frac{1}{r \\left(c - p\\right)} \\left(- c \\lambda p - c r + \\lambda p r + \\lambda r \\left(c - p\\right) + p r\\right)} \\right )}\\right)$$"
      ],
      "text/plain": [
       "                                              ⎛  -c⋅λ⋅p - c⋅r + λ⋅p⋅r + λ⋅r⋅(c\n",
       "                                              ⎜  ─────────────────────────────\n",
       "                                              ⎜                 r⋅(c - p)     \n",
       "c⋅λ⋅p + c⋅r - λ⋅p⋅r - p⋅r + r⋅(c - p)⋅LambertW⎝-ℯ                             \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                       λ⋅r⋅(c - p)                            \n",
       "\n",
       " - p) + p⋅r⎞\n",
       "───────────⎟\n",
       "           ⎟\n",
       "           ⎠\n",
       "────────────\n",
       "            "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol[0].simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is the result from Mathematica"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$f\\to \\frac{r (c-p) W\\left(-e^{\\frac{r (c (\\lambda -1)+p)-c \\lambda\n",
    "    p}{r (c-p)}}\\right)+c (\\lambda  p+r)-(\\lambda +1) p r}{\\lambda  r\n",
    "   (c-p)}$"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
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   "nbconvert_exporter": "python",
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