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 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": [],
     "source": [
      "> This is one of the 100 recipes of the [IPython Cookbook](http://ipython-books.github.io/), the definitive guide to high-performance scientific computing and data science in Python.\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# 15.3. Analyzing real-valued functions"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy import *\n",
      "init_printing()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "var('x z')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We define a new function depending on x."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f = 1/(1+x**2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's evaluate this function in 1."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f.subs(x, 1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can compute the derivative of this function..."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "diff(f, x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "limits..."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "limit(f, x, oo)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Taylor series..."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "series(f, x0=0, n=9)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Definite integrals..."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "integrate(f, (x, -oo, oo))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "indefinite integrals..."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "integrate(f, x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "and even Fourier transforms!"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fourier_transform(f, x, z)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "> You'll find all the explanations, figures, references, and much more in the book (to be released later this summer).\n",
      "\n",
      "> [IPython Cookbook](http://ipython-books.github.io/), by [Cyrille Rossant](http://cyrille.rossant.net), Packt Publishing, 2014 (500 pages)."
     ]
    }
   ],
   "metadata": {}
  }
 ]
}