Moffat-Derivatives.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Moffat2D"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "amplitude, x_0, y_0, alpha, gamma, x, y = sympy.symbols('amplitude x_0 y_0 alpha gamma x y')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\left(x - x_{0}\\right)^{2} + \\left(y - y_{0}\\right)^{2}}{\\gamma^{2}}$"
      ],
      "text/plain": [
       "((x - x_0)**2 + (y - y_0)**2)/gamma**2"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rr_gg = ((x - x_0) ** 2 + (y - y_0) ** 2) / gamma ** 2\n",
    "\n",
    "rr_gg"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle amplitude \\left(1 + \\frac{\\left(x - x_{0}\\right)^{2} + \\left(y - y_{0}\\right)^{2}}{\\gamma^{2}}\\right)^{- \\alpha}$"
      ],
      "text/plain": [
       "amplitude*(1 + ((x - x_0)**2 + (y - y_0)**2)/gamma**2)**(-alpha)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "moffat = amplitude * (1 + rr_gg) ** (-alpha)\n",
    "moffat"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "d_A = sympy.diff(moffat, amplitude)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(1 + \\frac{\\left(x - x_{0}\\right)^{2} + \\left(y - y_{0}\\right)^{2}}{\\gamma^{2}}\\right)^{- \\alpha}$"
      ],
      "text/plain": [
       "(1 + ((x - x_0)**2 + (y - y_0)**2)/gamma**2)**(-alpha)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d_A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d_A == (1 + rr_gg) ** (-alpha)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - amplitude \\left(1 + \\frac{\\left(x - x_{0}\\right)^{2} + \\left(y - y_{0}\\right)^{2}}{\\gamma^{2}}\\right)^{- \\alpha} \\log{\\left(1 + \\frac{\\left(x - x_{0}\\right)^{2} + \\left(y - y_{0}\\right)^{2}}{\\gamma^{2}} \\right)}$"
      ],
      "text/plain": [
       "-amplitude*(1 + ((x - x_0)**2 + (y - y_0)**2)/gamma**2)**(-alpha)*log(1 + ((x - x_0)**2 + (y - y_0)**2)/gamma**2)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d_alpha = sympy.diff(moffat, alpha)\n",
    "d_alpha"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d_alpha == -amplitude * d_A * sympy.log(1 + rr_gg)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{2 \\alpha amplitude \\left(1 + \\frac{\\left(x - x_{0}\\right)^{2} + \\left(y - y_{0}\\right)^{2}}{\\gamma^{2}}\\right)^{- \\alpha} \\left(\\left(x - x_{0}\\right)^{2} + \\left(y - y_{0}\\right)^{2}\\right)}{\\gamma^{3} \\left(1 + \\frac{\\left(x - x_{0}\\right)^{2} + \\left(y - y_{0}\\right)^{2}}{\\gamma^{2}}\\right)}$"
      ],
      "text/plain": [
       "2*alpha*amplitude*(1 + ((x - x_0)**2 + (y - y_0)**2)/gamma**2)**(-alpha)*((x - x_0)**2 + (y - y_0)**2)/(gamma**3*(1 + ((x - x_0)**2 + (y - y_0)**2)/gamma**2))"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d_gamma = sympy.diff(moffat, gamma)\n",
    "d_gamma"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# OLD VERSION\n",
    "d_gamma == 2 * amplitude * alpha * d_A * (rr_gg / (gamma * (1 + rr_gg)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{2 \\alpha amplitude \\left(1 + \\frac{\\left(x - x_{0}\\right)^{2} + \\left(y - y_{0}\\right)^{2}}{\\gamma^{2}}\\right)^{- \\alpha} \\left(\\left(x - x_{0}\\right)^{2} + \\left(y - y_{0}\\right)^{2}\\right)}{\\gamma^{3} \\left(1 + \\frac{\\left(x - x_{0}\\right)^{2} + \\left(y - y_{0}\\right)^{2}}{\\gamma^{2}}\\right)}$"
      ],
      "text/plain": [
       "2*alpha*amplitude*(1 + ((x - x_0)**2 + (y - y_0)**2)/gamma**2)**(-alpha)*((x - x_0)**2 + (y - y_0)**2)/(gamma**3*(1 + ((x - x_0)**2 + (y - y_0)**2)/gamma**2))"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2 * amplitude * alpha * d_A * (rr_gg / (gamma * (1 + rr_gg)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# NEW VERSION\n",
    "d_gamma == (2 * amplitude * alpha * d_A * rr_gg / (gamma ** 3 * (1 + rr_gg)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{2 \\alpha amplitude \\left(1 + \\frac{\\left(x - x_{0}\\right)^{2} + \\left(y - y_{0}\\right)^{2}}{\\gamma^{2}}\\right)^{- \\alpha} \\left(\\left(x - x_{0}\\right)^{2} + \\left(y - y_{0}\\right)^{2}\\right)}{\\gamma^{5} \\left(1 + \\frac{\\left(x - x_{0}\\right)^{2} + \\left(y - y_{0}\\right)^{2}}{\\gamma^{2}}\\right)}$"
      ],
      "text/plain": [
       "2*alpha*amplitude*(1 + ((x - x_0)**2 + (y - y_0)**2)/gamma**2)**(-alpha)*((x - x_0)**2 + (y - y_0)**2)/(gamma**5*(1 + ((x - x_0)**2 + (y - y_0)**2)/gamma**2))"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(2 * amplitude * alpha * d_A * rr_gg / (gamma ** 3 * (1 + rr_gg)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{\\alpha amplitude \\left(1 + \\frac{\\left(x - x_{0}\\right)^{2} + \\left(y - y_{0}\\right)^{2}}{\\gamma^{2}}\\right)^{- \\alpha} \\left(- 2 x + 2 x_{0}\\right)}{\\gamma^{2} \\left(1 + \\frac{\\left(x - x_{0}\\right)^{2} + \\left(y - y_{0}\\right)^{2}}{\\gamma^{2}}\\right)}$"
      ],
      "text/plain": [
       "-alpha*amplitude*(1 + ((x - x_0)**2 + (y - y_0)**2)/gamma**2)**(-alpha)*(-2*x + 2*x_0)/(gamma**2*(1 + ((x - x_0)**2 + (y - y_0)**2)/gamma**2))"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d_x_0 = sympy.diff(moffat, x_0)\n",
    "d_x_0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{2 \\alpha amplitude \\left(1 + \\frac{\\left(x - x_{0}\\right)^{2} + \\left(y - y_{0}\\right)^{2}}{\\gamma^{2}}\\right)^{- \\alpha} \\left(x - x_{0}\\right)}{\\gamma^{2} \\left(1 + \\frac{\\left(x - x_{0}\\right)^{2} + \\left(y - y_{0}\\right)^{2}}{\\gamma^{2}}\\right)}$"
      ],
      "text/plain": [
       "2*alpha*amplitude*(1 + ((x - x_0)**2 + (y - y_0)**2)/gamma**2)**(-alpha)*(x - x_0)/(gamma**2*(1 + ((x - x_0)**2 + (y - y_0)**2)/gamma**2))"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(2 * amplitude * alpha * d_A * (x - x_0) / (gamma ** 2 * (1 + rr_gg)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d_x_0 == (2 * amplitude * alpha * d_A * (x - x_0) / (gamma ** 2 * (1 + rr_gg)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Sympy is confused by 2 * (x - x_0) versus - (2 * x_0 - 2 * x)\n",
    "d_x_0 == ( - amplitude * alpha * d_A * (2 * x_0 - 2 * x) / (gamma ** 2 * (1 + rr_gg)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Moffat1D"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "amplitude, x_0, alpha, gamma, x = sympy.symbols('amplitude x_0 alpha gamma x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle amplitude \\left(1 + \\frac{\\left(x - x_{0}\\right)^{2}}{\\gamma^{2}}\\right)^{- \\alpha}$"
      ],
      "text/plain": [
       "amplitude*(1 + (x - x_0)**2/gamma**2)**(-alpha)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "moffat1d = amplitude * (1 + ((x - x_0) / gamma) ** 2) ** (-alpha)\n",
    "moffat1d"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "fac = (1 + (x - x_0) ** 2 / gamma ** 2)\n",
    "d_A = fac ** (-alpha)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{\\alpha amplitude \\left(1 + \\frac{\\left(x - x_{0}\\right)^{2}}{\\gamma^{2}}\\right)^{- \\alpha} \\left(- 2 x + 2 x_{0}\\right)}{\\gamma^{2} \\left(1 + \\frac{\\left(x - x_{0}\\right)^{2}}{\\gamma^{2}}\\right)}$"
      ],
      "text/plain": [
       "-alpha*amplitude*(1 + (x - x_0)**2/gamma**2)**(-alpha)*(-2*x + 2*x_0)/(gamma**2*(1 + (x - x_0)**2/gamma**2))"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d_x_0 = sympy.diff(moffat1d, x_0)\n",
    "d_x_0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# NEW\n",
    "d_x_0 == (2 * amplitude * alpha * (x - x_0) * d_A / (fac * gamma ** 2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{2 \\alpha amplitude \\left(1 + \\frac{\\left(x - x_{0}\\right)^{2}}{\\gamma^{2}}\\right)^{- \\alpha} \\left(x - x_{0}\\right)}{\\gamma^{2} \\left(1 + \\frac{\\left(x - x_{0}\\right)^{2}}{\\gamma^{2}}\\right)}$"
      ],
      "text/plain": [
       "2*alpha*amplitude*(1 + (x - x_0)**2/gamma**2)**(-alpha)*(x - x_0)/(gamma**2*(1 + (x - x_0)**2/gamma**2))"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(2 * amplitude * alpha * (x - x_0) * d_A / (fac * gamma ** 2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# OLD\n",
    "d_x_0 == (-amplitude * alpha * d_A * (-2 * x + 2 * x_0) / (gamma ** 2 * d_A ** alpha))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{\\alpha amplitude \\left(1 + \\frac{\\left(x - x_{0}\\right)^{2}}{\\gamma^{2}}\\right)^{- \\alpha} \\left(- 2 x + 2 x_{0}\\right) \\left(\\left(1 + \\frac{\\left(x - x_{0}\\right)^{2}}{\\gamma^{2}}\\right)^{- \\alpha}\\right)^{- \\alpha}}{\\gamma^{2}}$"
      ],
      "text/plain": [
       "-alpha*amplitude*(1 + (x - x_0)**2/gamma**2)**(-alpha)*(-2*x + 2*x_0)*((1 + (x - x_0)**2/gamma**2)**(-alpha))**(-alpha)/gamma**2"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(-amplitude * alpha * d_A * (-2 * x + 2 * x_0) / (gamma ** 2 * d_A ** alpha))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{2 \\alpha amplitude \\left(1 + \\frac{\\left(x - x_{0}\\right)^{2}}{\\gamma^{2}}\\right)^{- \\alpha} \\left(x - x_{0}\\right)^{2}}{\\gamma^{3} \\left(1 + \\frac{\\left(x - x_{0}\\right)^{2}}{\\gamma^{2}}\\right)}$"
      ],
      "text/plain": [
       "2*alpha*amplitude*(1 + (x - x_0)**2/gamma**2)**(-alpha)*(x - x_0)**2/(gamma**3*(1 + (x - x_0)**2/gamma**2))"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d_gamma = sympy.diff(moffat1d, gamma)\n",
    "d_gamma"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# OLD\n",
    "d_gamma == (2 * amplitude * alpha * d_A * (x - x_0) ** 2 / (gamma ** 3 * d_A ** alpha))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# NEW \n",
    "d_gamma == (2 * amplitude * alpha * (x - x_0) ** 2 * d_A / (fac * gamma ** 3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - amplitude \\left(1 + \\frac{\\left(x - x_{0}\\right)^{2}}{\\gamma^{2}}\\right)^{- \\alpha} \\log{\\left(1 + \\frac{\\left(x - x_{0}\\right)^{2}}{\\gamma^{2}} \\right)}$"
      ],
      "text/plain": [
       "-amplitude*(1 + (x - x_0)**2/gamma**2)**(-alpha)*log(1 + (x - x_0)**2/gamma**2)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d_alpha = sympy.diff(moffat1d, alpha)\n",
    "d_alpha"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# OLD\n",
    "d_alpha == -amplitude * d_A * sympy.log(1 + (x - x_0) ** 2 / gamma ** 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# NEW \n",
    "d_alpha == -amplitude * d_A * sympy.log(fac)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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