{ "nbformat": 2, "metadata": { "name": "sympy" }, "worksheets": [ { "cells": [ { "source": "

SymPy: Open Source Symbolic Mathematics

", "cell_type": "markdown" }, { "cell_type": "code", "language": "python", "outputs": [], "collapsed": true, "prompt_number": 1, "input": "%load_ext sympyprinting" }, { "cell_type": "code", "language": "python", "outputs": [], "collapsed": true, "prompt_number": 2, "input": "from __future__ import division\nfrom sympy import *\nx, y, z = symbols(\"x y z\")\nk, m, n = symbols(\"k m n\", integer=True)\nf, g, h = map(Function, 'fgh')" }, { "source": "

Elementary operations

", "cell_type": "markdown" }, { "cell_type": "code", "language": "python", "outputs": [ { "output_type": "pyout", "latex": "$$\\frac{3}{2} \\pi + \\frac{e^{\\mathbf{\\imath} x}}{x^{2} + y}$$", "prompt_number": 3, "png": 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"text": "\u2148\u22c5x \n3\u22c5\u03c0 \u212f \n\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\n 2 2 \n x + y" } ], "collapsed": false, "prompt_number": 3, "input": "Rational(3,2)*pi + exp(I*x) / (x**2 + y)\n" }, { "cell_type": "code", "language": "python", "outputs": [ { "output_type": "pyout", "latex": "$$-1.0$$", "prompt_number": 4, "png": "iVBORw0KGgoAAAANSUhEUgAAACsAAAASCAYAAADCKCelAAAABHNCSVQICAgIfAhkiAAAAV1JREFU\nSInt1b1LHEEYx/HP+VacTdKpKKKFXIqIIBZahBTX2Pk3pIpVIFhoMF26A8FKsLKzCqnTpxILUUjh\nW6FFbIIIuSKceBY7q8uyB9l7CSf4g+Hh+T47z/wYZmd4Qip0oOcX3KCSc14ZyzhBHRfYaq+1R5VE\nButYzTl3GlVMhrwfe/iQ/KinRYOxKlhBrcn5a/iB85DXsBv6jrbsroFK8u/sOG7xMcVfhV7vYtCu\nnW1FM+gVndWkTkN8G4NuMDsU4p8Ur4UxHINuMlvNqFXxIk66wezfEO8yav1J3g1mjxrwAor4FYO+\nRPE1tv37Q3GA9824S+kwxJcpXgxermKQNHuE+TYsnleXuMZYis+F+LDz//sYDGAigx9iMcXK+I2d\nTplZEF3kGw3q30UPwGyKvxH9+VMhH8Qx1jvg0Sfsi+7Kehg/g7mkKjjDSEaPJXzDJr7is+ixeNaz\nkroH5AA9SswlEREAAAAASUVORK5CYII=\n", "text": "-1.00000000000000" } ], "collapsed": false, "prompt_number": 4, "input": "exp(I*x).subs(x,pi).evalf()\n" }, { "cell_type": "code", "language": "python", "outputs": [], "collapsed": true, "prompt_number": 5, "input": "e = x + 2*y" }, { "cell_type": "code", "language": "python", "outputs": [ { "output_type": "pyout", "prompt_number": 6, "text": "Add(Symbol('x'), Mul(Integer(2), Symbol('y')))" } ], "collapsed": false, "prompt_number": 6, "input": "srepr(e)" }, { "cell_type": "code", "language": "python", "outputs": [ { "output_type": "pyout", "latex": "$$262537412640768743.99999999999925007259719818568888$$", "prompt_number": 7, "png": "iVBORw0KGgoAAAANSUhEUgAAAfwAAAASCAYAAAC+a2xVAAAABHNCSVQICAgIfAhkiAAACXtJREFU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\n", "text": "262537412640768743.99999999999925007259719818568888" } ], "collapsed": false, "prompt_number": 7, "input": "exp(pi * sqrt(163)).evalf(50)" }, { "source": "

Algebra

", "cell_type": "markdown" }, { "cell_type": "code", "language": "python", "outputs": [ { "output_type": "pyout", "latex": "$$x^{3} + 2 x^{2} y + x^{2} + x y^{2} + 2 x y + y^{2}$$", "prompt_number": 8, "png": "iVBORw0KGgoAAAANSUhEUgAAAQ0AAAAbCAYAAABm6to6AAAABHNCSVQICAgIfAhkiAAABXNJREFU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\nOFM1l1Xqnqo5TyQSiUQikUgkEv8v/gNI9gkr+UdTSgAAAABJRU5ErkJggg==\n", "text": "3 2 2 2 2\nx + 2\u22c5x \u22c5y + x + x\u22c5y + 2\u22c5x\u22c5y + y" } ], "collapsed": false, "prompt_number": 8, "input": "((x+y)**2 * (x+1)).expand()" }, { "cell_type": "code", "language": "python", "outputs": [ { "output_type": "display_data", "latex": "$$\\frac{x \\operatorname{sin}\\left(x\\right) -1}{x} + \\frac{1}{x}$$", "png": "iVBORw0KGgoAAAANSUhEUgAAAHUAAAAeCAYAAAASN1ElAAAABHNCSVQICAgIfAhkiAAAAyFJREFU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\noZik/6S2SgAAAABJRU5ErkJggg==\n", "text": "x\u22c5sin(x) - 1 1\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\n x x" }, { "output_type": "pyout", "latex": "$$\\operatorname{sin}\\left(x\\right)$$", "prompt_number": 9, "png": "iVBORw0KGgoAAAANSUhEUgAAADAAAAASCAYAAAAdZl26AAAABHNCSVQICAgIfAhkiAAAAnVJREFU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\nfj0LzBdrAAAAAElFTkSuQmCC\n", "text": "sin(x)" } ], "collapsed": false, "prompt_number": 9, "input": "a = 1/x + (x*sin(x) - 1)/x\ndisplay(a)\nsimplify(a)" }, { "cell_type": "code", "language": "python", "outputs": [ { "output_type": "pyout", "prompt_number": 10, "text": "[-2\u22c5\u2148, 2\u22c5\u2148, -2]" } ], "collapsed": false, "prompt_number": 10, "input": "solve(Eq(x**3 + 2*x**2 + 4*x + 8, 0), x)" }, { "cell_type": "code", "language": "python", "outputs": [ { "output_type": "pyout", "latex": "$$\\sum_{n=a}^{b} \\left(2^{n} + 6 n^{2}\\right)$$", "prompt_number": 11, "png": "iVBORw0KGgoAAAANSUhEUgAAAHgAAAA5CAYAAAD9YO8bAAAABHNCSVQICAgIfAhkiAAAB5NJREFU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\nCnsQh3MAAAAASUVORK5CYII=\n", "text": "b \n ___ \n \\ ` \n \\ \u239b n 2\u239e\n / \u239d2 + 6\u22c5n \u23a0\n /__, \nn = a" } ], "collapsed": false, "prompt_number": 11, "input": "a, b = symbols('a b')\nSum(6*n**2 + 2**n, (n, a, b))" }, { "source": "

Calculus

", "cell_type": "markdown" }, { "cell_type": "code", "language": "python", "outputs": [ { "output_type": "pyout", "latex": "$$- \\frac{1}{6}$$", "prompt_number": 12, "png": "iVBORw0KGgoAAAANSUhEUgAAACYAAAAiCAYAAAAzrKu4AAAABHNCSVQICAgIfAhkiAAAASpJREFU\nWIXt178rRlEYwPEPXikZLH4NShTKoAyymw3KPyJlsBssVqvBYDIwCcl/gHpXr8liEYWN4bx0k3Ku\nzsmr937rDGd47vPtuc+5zz20OYN/LfCVUWxjp2xgLb3LJzOYxxCeygbnFKs31xT6ywZ3JtdJRCVW\nlrYW62iuUnRlEPlgHOtYwLBwOu9wnzFnRetRuim/4S3BM/4PxYpNYlV8FRvYSm7UpCjRg2nxYg+4\nTS1U8VtSnMqfmBP6tw97eI0Jyj0rNzGCfQxgLXO+KBZxUNgvC3Mzipyvchc3uMYYrnCeMV80dZyi\nVyjAMVZig3P2WDcu8SyMrRNsxAbnFGvgsbB/Ef7LosgpdoiJwn4YFxnzRVPDEZaEi++ZcPltGWaF\nz0TLXnwq2pN3yugjnSzSytcAAAAASUVORK5CYII=\n", "text": "-1/6" } ], "collapsed": false, "prompt_number": 12, "input": "limit((sin(x)-x)/x**3, x, 0)" }, { "cell_type": "code", "language": "python", "outputs": [ { "output_type": "pyout", "latex": "$$1 + \\frac{1}{2} x^{2} + \\frac{5}{24} x^{4} + \\operatorname{\\mathcal{O}}\\left(x^{6}\\right)$$", "prompt_number": 13, "png": "iVBORw0KGgoAAAANSUhEUgAAANoAAAAiCAYAAAA53MevAAAABHNCSVQICAgIfAhkiAAABuNJREFU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\nAAAAAElFTkSuQmCC\n", "text": "2 4 \n x 5\u22c5x \n1 + \u2500\u2500 + \u2500\u2500\u2500\u2500 + O(x**6)\n 2 24" } ], "collapsed": false, "prompt_number": 13, "input": "(1/cos(x)).series(x, 0, 6)" }, { "cell_type": "code", "language": "python", "outputs": [ { "output_type": "pyout", "latex": "$$- 4 \\frac{x \\operatorname{sin}\\left(x^{2}\\right) \\operatorname{cos}\\left(x^{2}\\right)}{x + 1} - \\frac{\\operatorname{cos}^{2}\\left(x^{2}\\right)}{\\left(x + 1\\right)^{2}}$$", "prompt_number": 14, "png": 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"text": "2 \n \u239b 2\u239e \u239b 2\u239e \u239b 2\u239e\n 4\u22c5x\u22c5sin\u239dx \u23a0\u22c5cos\u239dx \u23a0 cos \u239dx \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\n x + 1 2\n (x + 1)" } ], "collapsed": false, "prompt_number": 14, "input": "diff(cos(x**2)**2 / (1+x), x)" }, { "cell_type": "code", "language": "python", "outputs": [ { "output_type": "pyout", "latex": "$$-2 + \\frac{1}{4} \\pi^{2}$$", "prompt_number": 15, "png": 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"text": "2\n \u03c0 \n-2 + \u2500\u2500\n 4" } ], "collapsed": false, "prompt_number": 15, "input": "integrate(x**2 * cos(x), (x, 0, pi/2))" }, { "cell_type": "code", "language": "python", "outputs": [ { "output_type": "display_data", "latex": "$$9 \\operatorname{f}\\left(x\\right) + \\frac{\\partial^{2}}{\\partial^{2} x} \\operatorname{f}\\left(x\\right) = 1$$", "png": 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"text": "2 \n d \n9\u22c5f(x) + \u2500\u2500\u2500(f(x)) = 1\n 2 \n dx" }, { "output_type": "pyout", "latex": "$$\\operatorname{f}\\left(x\\right) = C_{1} \\operatorname{cos}\\left(3 x\\right) + C_{2} \\operatorname{sin}\\left(3 x\\right) + \\frac{1}{9}$$", "prompt_number": 16, "png": 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"text": "f(x) = C\u2081\u22c5cos(3\u22c5x) + C\u2082\u22c5sin(3\u22c5x) + 1/9" } ], "collapsed": false, "prompt_number": 16, "input": "eqn = Eq(Derivative(f(x),x,x) + 9*f(x), 1)\ndisplay(eqn)\ndsolve(eqn, f(x))" } ] } ] }