Doodling on stationary approximation

Stationary approx

{
 "metadata": {
  "name": "stationary_phase"
 }, 
 "name": "stationary_phase", 
 "nbformat": 2, 
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown", 
     "source": "This notebook is a result of my attempts to understand stationary phase approximation. I was trying to come\nup with a problem where the approach could be used, but I wanted to keep matters simpler than in class and so I\ndropped as many parameters as possible. As a result I considered integrals of the\nform $\\eta = \\int_{-\\infty}^{\\infty}f(k)exp(g(k))dk$ , with $f$ and $g$ real functions. Note that $\\eta$ is not a\nfunction but a number. If $g$ were complex, one could perhaps view $\\eta$ as the value of surface elevation comming\nfrom the Fourier integral evaluated at some fixed time $t$ and point $x$. But the point of this exercise was not\nto come up with physically meaningful examples. \n\nAn example of integral where stationary approximation works extremely well is\n$I = \\int_{-\\infty}^{\\infty}exp(-k^2)dk$. This is easy to see and while boring it is a good starting point. So \nlet's complicate things a bit. Putting the physical interpretation aside consider,\n$\\eta=\\int_{-\\infty}^{\\infty}cos(k)exp(-k-k^2)dx$ . The integrand is plotted below"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "%pylab inline\n\nimport numpy as np\n\nk = np.linspace(-4, 4, 100)\ny = np.exp(-k**2)*cos(k)\nplot(k, y)", 
     "language": "python", 
     "outputs": [
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       "stream": "stdout", 
       "text": "\nWelcome to pylab, a matplotlib-based Python environment [backend: module://IPython.zmq.pylab.backend_inline].\nFor more information, type 'help(pylab)'."
      }, 
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       "output_type": "pyout", 
       "prompt_number": 1, 
       "text": "[<matplotlib.lines.Line2D at 0xb2b51ac>]"
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       "output_type": "display_data", 
       "png": 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btgxZWVk4cOAAJk6ciGXLlrVZxmKx4E9/+hP27duHoqIi/PnPf0Z5eblPgYmI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\nS9oAAAAASUVORK5CYII=\n"
      }
     ], 
     "prompt_number": 1
    }, 
    {
     "cell_type": "markdown", 
     "source": "The integral can be computed and the result is quite an impressive expression"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "import sympy as sym\n\n%load_ext sympyprinting\n\nk = sym.Symbol('k')\nf = sym.exp(-k-k**2)*sym.cos(k)\nexact =sym.integrate(f, (k, -sym.oo, sym.oo))\n\nexact", 
     "language": "python", 
     "outputs": [
      {
       "latex": "$$\\frac{\\frac{\\pi^{\\frac{3}{2}} \\lvert{\\left(\\mathbf{\\imath} \\sin{\\left (\\frac{1}{2} \\right )} S\\left(\\frac{1}{\\sqrt{\\pi}}\\right) + \\mathbf{\\imath} \\cos{\\left (\\frac{1}{2} \\right )} C\\left(\\frac{1}{\\sqrt{\\pi}}\\right)\\right) e^{- \\frac{1}{4} \\mathbf{\\imath} \\pi}}\\rvert \\Gamma\\left(- \\frac{1}{4}\\right)}{4 \\Gamma\\left(\\frac{3}{4}\\right)} - \\frac{3}{16} \\frac{\\pi^{\\frac{3}{2}} \\left(- \\cos{\\left (\\frac{1}{2} \\right )} S\\left(\\frac{1}{\\sqrt{\\pi}}\\right) + \\sin{\\left (\\frac{1}{2} \\right )} C\\left(\\frac{1}{\\sqrt{\\pi}}\\right)\\right) \\Gamma\\left(- \\frac{3}{4}\\right)}{\\Gamma\\left(\\frac{5}{4}\\right)} + \\frac{\\cos{\\left (\\frac{1}{2} \\right )} \\Gamma^{2}\\left(\\frac{1}{4}\\right) \\Gamma^{2}\\left(\\frac{3}{4}\\right)}{2 \\sqrt{\\pi}}}{2 \\pi} - \\frac{- \\frac{\\cos{\\left (\\frac{1}{2} \\right )} \\Gamma^{2}\\left(\\frac{1}{4}\\right) \\Gamma^{2}\\left(\\frac{3}{4}\\right)}{2 \\sqrt{\\pi}} + \\frac{\\mathbf{\\imath} \\pi^{\\frac{3}{2}} \\left(- \\mathbf{\\imath} \\cos{\\left (\\frac{1}{2} \\right )} C\\left(\\frac{1}{\\sqrt{\\pi}}\\right) - \\mathbf{\\imath} \\sin{\\left (\\frac{1}{2} \\right )} S\\left(\\frac{1}{\\sqrt{\\pi}}\\right)\\right) \\Gamma\\left(- \\frac{1}{4}\\right)}{4 \\Gamma\\left(\\frac{3}{4}\\right)} - \\frac{3}{16} \\frac{\\pi^{\\frac{3}{2}} \\left(- \\cos{\\left (\\frac{1}{2} \\right )} S\\left(\\frac{1}{\\sqrt{\\pi}}\\right) + \\sin{\\left (\\frac{1}{2} \\right )} C\\left(\\frac{1}{\\sqrt{\\pi}}\\right)\\right) \\Gamma\\left(- \\frac{3}{4}\\right)}{\\Gamma\\left(\\frac{5}{4}\\right)}}{2 \\pi}$$", 
       "output_type": "pyout", 
       "prompt_number": 2, 
       "text": "\n     \u2502                                                         -\u2148\u22c5\u03c0\u2502          \n     \u2502                                                         \u2500\u2500\u2500\u2500\u2502          \n 3/2 \u2502\u239b                   \u239b 1  \u239e                      \u239b 1  \u239e\u239e   4  \u2502          \n\u03c0   \u22c5\u2502\u239c\u2148\u22c5sin(1/2)\u22c5fresnels\u239c\u2500\u2500\u2500\u2500\u239f + \u2148\u22c5cos(1/2)\u22c5fresnelc\u239c\u2500\u2500\u2500\u2500\u239f\u239f\u22c5\u212f    \u2502\u22c5\u0393(-1/4)  \n     \u2502\u239c                   \u239c 1/2\u239f                      \u239c 1/2\u239f\u239f      \u2502          \n     \u2502\u239d                   \u239d\u03c0   \u23a0                      \u239d\u03c0   \u23a0\u23a0      \u2502          \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 -\n                                  4\u22c5\u0393(3/4)                                    \n                                                                              \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n\n                                                                              \n                                                                              \n    3/2 \u239b                   \u239b 1  \u239e                    \u239b 1  \u239e\u239e                 \n 3\u22c5\u03c0   \u22c5\u239c- cos(1/2)\u22c5fresnels\u239c\u2500\u2500\u2500\u2500\u239f + sin(1/2)\u22c5fresnelc\u239c\u2500\u2500\u2500\u2500\u239f\u239f\u22c5\u0393(-3/4)         \n        \u239c                   \u239c 1/2\u239f                    \u239c 1/2\u239f\u239f                 \n        \u239d                   \u239d\u03c0   \u23a0                    \u239d\u03c0   \u23a0\u23a0           cos(1/\n \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\n                              16\u22c5\u0393(5/4)                                       \n                                                                              \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n       2\u22c5\u03c0                                                                    \n\n                                                                              \n                                                                              \n                                                     3/2 \u239b                    \n                                                  \u2148\u22c5\u03c0   \u22c5\u239c- \u2148\u22c5cos(1/2)\u22c5fresnel\n         2       2                    2       2          \u239c                    \n2)\u22c5\u0393(1/4) \u22c5\u0393(3/4)      cos(1/2)\u22c5\u0393(1/4) \u22c5\u0393(3/4)           \u239d                    \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500   - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n      ___                          ___                                        \n  2\u22c5\u2572\u2571 \u03c0                       2\u22c5\u2572\u2571 \u03c0                                         \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n\n                                                                              \n                                                                              \n \u239b 1  \u239e                      \u239b 1  \u239e\u239e              3/2 \u239b                   \u239b 1 \nc\u239c\u2500\u2500\u2500\u2500\u239f - \u2148\u22c5sin(1/2)\u22c5fresnels\u239c\u2500\u2500\u2500\u2500\u239f\u239f\u22c5\u0393(-1/4)   3\u22c5\u03c0   \u22c5\u239c- cos(1/2)\u22c5fresnels\u239c\u2500\u2500\u2500\n \u239c 1/2\u239f                      \u239c 1/2\u239f\u239f                  \u239c                   \u239c 1/\n \u239d\u03c0   \u23a0                      \u239d\u03c0   \u23a0\u23a0                  \u239d                   \u239d\u03c0  \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n    4\u22c5\u0393(3/4)                                                                16\n                                                                              \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                           2\u22c5\u03c0                                                \n\n                                     \n                                     \n \u239e                    \u239b 1  \u239e\u239e        \n\u2500\u239f + sin(1/2)\u22c5fresnelc\u239c\u2500\u2500\u2500\u2500\u239f\u239f\u22c5\u0393(-3/4)\n2\u239f                    \u239c 1/2\u239f\u239f        \n \u23a0                    \u239d\u03c0   \u23a0\u23a0        \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u22c5\u0393(5/4)                              \n                                     \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                     "
      }
     ], 
     "prompt_number": 2
    }, 
    {
     "cell_type": "markdown", 
     "source": "Evaluating the expression gives even more impressive(in its simplicity) number $cos(\\frac{1}{2})\\sqrt{\\pi}$ .\nTo see this"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "print exact.evalf()", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "stream", 
       "stream": "stdout", 
       "text": "1.55547459131012"
      }
     ], 
     "prompt_number": 3
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "exact_eval = sym.cos(0.5)*sym.sqrt(pi)\nexact_eval", 
     "language": "python", 
     "outputs": [
      {
       "latex": "$$1.55547459131012$$", 
       "output_type": "pyout", 
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJ0AAAASCAYAAABBwNzbAAAABHNCSVQICAgIfAhkiAAABTdJREFU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\n35qUPnulS20LLbTQQgsttNBc/A1/5MzdfH7RmgAAAABJRU5ErkJggg==\n", 
       "prompt_number": 4, 
       "text": "1.55547459131012"
      }
     ], 
     "prompt_number": 4
    }, 
    {
     "cell_type": "markdown", 
     "source": "Now comes the stationary approximation. We want to do a Taylor expansion of the integral around $k_0$ which is\na point where the derivative of $\\chi(k)=-k-k^2$ is zero. This gives $k_0 = -\\frac{1}{2}$. Moreover, the Taylor\nexpansion around $k_0$ just gives the same polynomial as $\\chi$, $\\chi(k)=\\frac{1}{4}-(k+\\frac{1}{2})^2=-k-k^2$.\nThe stationary phase approximations than gives\n$\\eta\\approx cos(k_0)\\int_{-\\infty}^{\\infty}exp(\\frac{1}{4}-(k+\\frac{1}{2})^2)$ dk. Cosine is even so $cos(-0.5)=cos(0.5)$.\nThe integral gives "
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "approx = sym.integrate(sym.exp(0.25-(k+0.5)**2), (k, -sym.oo, sym.oo))\napprox", 
     "language": "python", 
     "outputs": [
      {
       "latex": "$$1.28402541668774 \\sqrt{\\pi}$$", 
       "output_type": "pyout", 
       "png": "iVBORw0KGgoAAAANSUhEUgAAALsAAAAWCAYAAACYEu1aAAAABHNCSVQICAgIfAhkiAAABn1JREFU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\n/w9/A5BjVF1hGW7aAAAAAElFTkSuQmCC\n", 
       "prompt_number": 7, 
       "text": "\n                   ___\n1.28402541668774\u22c5\u2572\u2571 \u03c0 "
      }
     ], 
     "prompt_number": 7
    }, 
    {
     "cell_type": "markdown", 
     "source": "So finally "
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "(approx*sym.cos(0.5).evalf()).evalf()", 
     "language": "python", 
     "outputs": [
      {
       "latex": "$$1.99726891025417$$", 
       "output_type": "pyout", 
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJ4AAAASCAYAAACq92fYAAAABHNCSVQICAgIfAhkiAAABhdJREFU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\nw/02gxnMYAYzmMEM/rf4L2pqBt+WoDG5AAAAAElFTkSuQmCC\n", 
       "prompt_number": 11, 
       "text": "1.99726891025417"
      }
     ], 
     "prompt_number": 11
    }, 
    {
     "cell_type": "markdown", 
     "source": "Close enough? Is this not suitable for the approximation? Figure out better example ... "
    }, 
    {
     "cell_type": "code", 
     "collapsed": true, 
     "input": "", 
     "language": "python", 
     "outputs": []
    }
   ]
  }
 ]
}