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+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Chapter 22 : Integral Transform"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 22.1, page no. 608"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 6,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "To find the fourier sin integral\n",
+ "0.636619772367581*Integral(sin(t*u), (t, 0, oo))*Integral(sin(u*x), (u, 0, oo))\n"
+ ]
+ }
+ ],
+ "source": [
+ "import sympy,math\n",
+ "\n",
+ "print \"To find the fourier sin integral\"\n",
+ "x = sympy.Symbol('x')\n",
+ "t = sympy.Symbol('t')\n",
+ "u = sympy.Symbol('u')\n",
+ "fs = 2/math.pi*sympy.integrate(sympy.sin(u*x),(u,0,sympy.oo))*(sympy.integrate(x**0*sympy.sin(u*t),(t,0,sympy.oo)))\n",
+ "print fs"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 22.2, page no. 608"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 8,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "To find the fourier transform of given function\n",
+ "Piecewise((2, s == 0), (1.0*I*exp(-1.0*I*s)/s - 1.0*I*exp(1.0*I*s)/s, True))\n",
+ "pi/2\n"
+ ]
+ }
+ ],
+ "source": [
+ "import sympy\n",
+ "\n",
+ "print \"To find the fourier transform of given function\"\n",
+ "x = sympy.Symbol('x')\n",
+ "s = sympy.Symbol('s')\n",
+ "F = sympy.integrate(sympy.exp(1j*s*x),(x,-1,1))\n",
+ "print F\n",
+ "F1 = sympy.integrate(sympy.sin(x)/x,(x,0,sympy.oo))\n",
+ "print F1"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {
+ "collapsed": true
+ },
+ "source": [
+ "## Example 22.3, page no. 609"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 8,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "To find the fourier transform of given function\n",
+ "Piecewise((4/3, s**6 == 0), ((-2.0*s**4 - 2.0*I*s**3)*exp(1.0*I*s)/s**6 - (2.0*s**4 - 2.0*I*s**3)*exp(-1.0*I*s)/s**6, True))\n",
+ "Integral((x*cos(x) - sin(x))*cos(x/2)/x**3, (x, 0, +inf))\n"
+ ]
+ }
+ ],
+ "source": [
+ "import sympy,numpy\n",
+ "\n",
+ "print \"To find the fourier transform of given function\"\n",
+ "x = sympy.Symbol('x')\n",
+ "s = sympy.Symbol('s')\n",
+ "F = sympy.integrate(sympy.exp(1j*s*x)*(1-x**2),(x,-1,1))\n",
+ "print F\n",
+ "F1 = sympy.integrals.Integral((x*sympy.cos(x)-sympy.sin(x))/x**3*sympy.cos(x/2),(x,0,numpy.inf))\n",
+ "print F1"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 22.4, page no. 610"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 10,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "To find the fourier sin integral\n",
+ "Piecewise((s/(s**2 + 1), Abs(periodic_argument(polar_lift(s)**2, oo)) == 0), (Integral(exp(-x)*sin(s*x), (x, 0, oo)), True))\n",
+ "Piecewise((sqrt(pi)*(-sqrt(pi)*sinh(m) + sqrt(pi)*cosh(m))/2, Abs(periodic_argument(polar_lift(m)**2, oo)) == 0), (Integral(x*sin(m*x)/(x**2 + 1), (x, 0, oo)), True))\n"
+ ]
+ }
+ ],
+ "source": [
+ "import sympy,math\n",
+ "\n",
+ "print \"To find the fourier sin integral\"\n",
+ "x = sympy.Symbol('x')\n",
+ "s = sympy.Symbol('s')\n",
+ "m = sympy.Symbol('m')\n",
+ "fs = sympy.integrate(sympy.sin(s*x)*sympy.exp(-x),(x,0,sympy.oo))\n",
+ "print fs\n",
+ "f = sympy.integrate(x*sympy.sin(m*x)/(1+x**2),(x,0,sympy.oo))\n",
+ "print f"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 22.5, page no. 611"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 11,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "Fourier cosine transform.\n",
+ "Piecewise((1/2, s == 0), (-sin(s)/s + cos(s)/s**2 - cos(2*s)/s**2, True)) + Piecewise((1/2, s == 0), (sin(s)/s + cos(s)/s**2 - 1/s**2, True))\n"
+ ]
+ }
+ ],
+ "source": [
+ "import sympy,math\n",
+ "\n",
+ "print \"Fourier cosine transform.\"\n",
+ "x = sympy.Symbol('x')\n",
+ "s = sympy.Symbol('s')\n",
+ "f = sympy.integrate(x*sympy.cos(s*x),(x,0,1))+sympy.integrate((2-x)*sympy.cos(s*x),(x,1,2))\n",
+ "print f"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 22.6, page no. 612"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 12,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "Fourier cosine transform.\n",
+ "Piecewise((s*atan(sqrt(s**2/a**2))/(a*sqrt(s**2/a**2)), Or(And(-s**2/a**2 != 1, Abs(periodic_argument(polar_lift(a)**2, oo)) == pi, Abs(periodic_argument(polar_lift(s)**2, oo)) == 0), And(Abs(periodic_argument(polar_lift(a)**2, oo)) < pi, Abs(periodic_argument(polar_lift(s)**2, oo)) == 0))), (Integral(exp(-a*x)*sin(s*x)/x, (x, 0, oo)), True))\n"
+ ]
+ }
+ ],
+ "source": [
+ "import sympy,math\n",
+ "\n",
+ "print \"Fourier cosine transform.\"\n",
+ "x = sympy.Symbol('x')\n",
+ "s = sympy.Symbol('s')\n",
+ "a = sympy.Symbol('a')\n",
+ "f = sympy.integrate(sympy.exp(-a*x)/x*sympy.sin(s*x),(x,0,sympy.oo))\n",
+ "print f"
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 2",
+ "language": "python",
+ "name": "python2"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 2
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython2",
+ "version": "2.7.10"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 0
+}
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