{ "cells": [ { "cell_type": "markdown", "id": "07221538", "metadata": {}, "source": [ "\\begin{equation}\n", "I=\\int_0^{\\infty}\\dfrac{x^{2/3}}{e^x-e^{-x}}dx\n", "\\end{equation}\n", "\n", "Esta integral tiene dos problemas, se indetermina en cero y va hasta infinito. Primero, soluciones el problema del cero con el siguiente cambio de variable\n", "\n", "\\begin{equation}\n", "x=y^2\\\\\n", "dx=2ydy\n", "\\end{equation}\n", "transformando la integral en\n", "\\begin{equation}\n", "I=2\\int_0^{\\infty}\\dfrac{y^{7/3}}{e^{(y^2)}-e^{(-y^2)}}dy\n", "\\end{equation}\n", "Esta integral tiende a cero cuando y tiende a cero, por lo tanto no hay problema en el limite inferior. Para arreglar el limite superior vamos a separar la integral en dos.\n", "\\begin{equation}\n", "I=I_1+I_2\\\\\n", "I_1=2\\int_0^{1}\\dfrac{y^{7/3}}{e^{(y^2)}-e^{(-y^2)}}dy =\\int_0^{1} f(y)dy \\\\\n", "I_2=2\\int_1^{\\infty}\\dfrac{y^{7/3}}{e^{(y^2)}-e^{(-y^2)}}dy\n", "\\end{equation}\n", "\n", "Ahora solo debemos trabajar en $I_2$, procedemos aplicando este nuevo cambio de variable\n", "\\begin{equation}\n", "u=e^{-y}\\\\\n", "-2\\dfrac{\\ln(u)}{u}du=dy\n", "\\end{equation}\n", "\n", "lo que nos entrega\n", "\\begin{equation}\n", "I_2=\\int_0^{e^-1} \\dfrac{\\ln^{7/3}(u^{-1})}{u \\sinh(\\ln^2(u))}du =\\int_0^{e^-1} g(u)du\n", "\\end{equation}" ] }, { "cell_type": "code", "execution_count": 62, "id": "7c6997fc", "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "import scipy\n", "from scipy.integrate import romberg\n", "from scipy.integrate import quad\n", "from scipy.integrate import trapezoid\n", "from scipy.integrate import simpson\n", "import matplotlib.pyplot as plt" ] }, { "cell_type": "code", "execution_count": 63, "id": "833a7735", "metadata": {}, "outputs": [], "source": [ "def I(x):\n", " return x**(2./3.)/(np.exp(x)-np.exp(-x))\n", "def I1(x):\n", " if x==0: \n", " return 0\n", " else:\n", " return 2.*x**(7./3.)/(np.exp(x**2.)-np.exp(-x**2.))\n", "def I2(x):\n", " if x==0:\n", " return 0\n", " else:\n", " return np.log(1./x)**(7./3.)/(x*np.sinh(np.log(x)**2.))" ] }, { "cell_type": "code", "execution_count": 88, "id": "56dce091", "metadata": { "scrolled": true }, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "N=np.logspace(0.3,5,15)\n", "simpN=np.empty(len(N))\n", "for j in range(len(N)):\n", " y=np.linspace(0,1,int(N[j]))\n", " fy=np.empty(len(y))\n", " for i in range(len(y)):\n", " fy[i]=I1(y[i])\n", " u=np.linspace(0,np.exp(-1),int(N[j]))\n", " gu=np.empty(len(u))\n", " for i in range(len(u)):\n", " gu[i]=I2(u[i])\n", " simpN[j]=simpson(gu,u)+simpson(fy,y)\n", "plt.loglog(N,simpN) \n", "plt.xlabel('N')\n", "plt.ylabel(r'I=$\\int_0^{\\infty} (...)dx$')\n", "#plt.plot(y,fy,label=r'I$_1$')\n", "#plt.plot(u,gu,label=r'I$_2$')\n", "#plt.legend(loc='best',fontsize=15)\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 70, "id": "d2e70dd0", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1.3131830448037718\n" ] } ], "source": [ "print(simpN[-1])" ] }, { "cell_type": "code", "execution_count": 65, "id": "e765a947", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "I1=0.7200830602033684\n", "I2=0.5924872129527814\n", "I=1.3125702731561497\n" ] } ], "source": [ "Trap1=trapezoid(fy,y)\n", "print('I1='+str(Trap1))\n", "Trap2=trapezoid(gu,u)\n", "print('I2='+str(Trap2))\n", "print('I='+str(Trap1+Trap2))" ] }, { "cell_type": "code", "execution_count": 66, "id": "9efe65ab", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "I1=0.7202670641833189\n", "I2=0.5924922975635774\n", "I=1.3127593617468962\n" ] } ], "source": [ "Simp1=simpson(fy,y)\n", "print('I1='+str(Simp1))\n", "Simp2=simpson(gu,u)\n", "print('I2='+str(Simp2))\n", "print('I='+str(Simp1+Simp2))" ] }, { "cell_type": "code", "execution_count": 71, "id": "aeb7a75e", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1.3131830866025627 3.6438660977466952e-09\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ ":2: RuntimeWarning: overflow encountered in exp\n", " return x**(2./3.)/(np.exp(x)-np.exp(-x))\n" ] } ], "source": [ "res,err=quad(I,0,np.inf)\n", "print(res,err)" ] }, { "cell_type": "code", "execution_count": 75, "id": "d37c64f0", "metadata": {}, "outputs": [], "source": [ "import cmath\n", "\n", "def complex_quad(func, a, b, **kwargs):\n", " def real_func(x):\n", " return scipy.real(func(x))\n", " def imag_func(x):\n", " return scipy.imag(func(x))\n", " real_integral = quad(real_func, a, b, **kwargs)\n", " imag_integral = quad(imag_func, a, b, **kwargs)\n", " return (real_integral[0] + 1j*imag_integral[0], real_integral[1:], imag_integral[1:])" ] }, { "cell_type": "markdown", "id": "c23f8b84", "metadata": {}, "source": [ "\\begin{equation}\n", "\\oint \\dfrac{\\cos(z)}{z}dz\n", "\\end{equation}\n", "C.v.\n", "\n", "\\begin{equation}\n", "z=e^{i\\theta}\\\\\n", "dz=ie^{i\\theta}d\\theta\n", "\\end{equation}\n", "\n", "\\begin{equation}\n", "\\int_0^{2\\pi} i\\cos(e^{i\\theta})d\\theta\n", "\\end{equation}" ] }, { "cell_type": "code", "execution_count": 76, "id": "ed697887", "metadata": {}, "outputs": [], "source": [ "def z(t):\n", " #return 1\n", " return 1j*cmath.cos(cmath.exp(1j*t))" ] }, { "cell_type": "code", "execution_count": 77, "id": "1244f4d9", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "(-9.00480655960687e-17+6.283185307179585j)\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ ":5: DeprecationWarning: scipy.real is deprecated and will be removed in SciPy 2.0.0, use numpy.real instead\n", " return scipy.real(func(x))\n", ":7: DeprecationWarning: scipy.imag is deprecated and will be removed in SciPy 2.0.0, use numpy.imag instead\n", " return scipy.imag(func(x))\n" ] } ], "source": [ "res=complex_quad(z,0,2*np.pi)\n", "print(res[0])" ] }, { "cell_type": "code", "execution_count": 74, "id": "980761ce", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(5+3.141592653589793j)" ] }, "execution_count": 74, "metadata": {}, "output_type": "execute_result" } ], "source": [ "5+np.pi*1j" ] }, { "cell_type": "code", "execution_count": null, "id": "696da607", "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "id": "192e7798", "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "id": "2d352f34", "metadata": {}, "outputs": [], "source": [] } ], "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", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.8" } }, "nbformat": 4, "nbformat_minor": 5 }