{ "cells": [ { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "# Un premier contact avec SageMath\n", "\n", "On essaye les maths avec $\\pi$, ..." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "4" ] }, "execution_count": 1, "metadata": { }, "output_type": "execute_result" } ], "source": [ "2+2" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "-1" ] }, "execution_count": 2, "metadata": { }, "output_type": "execute_result" } ], "source": [ "i*i" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "-1" ] }, "execution_count": 4, "metadata": { }, "output_type": "execute_result" } ], "source": [ "i**2" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "-1" ] }, "execution_count": 5, "metadata": { }, "output_type": "execute_result" } ], "source": [ "i^2" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "'i**Integer(2)'" ] }, "execution_count": 6, "metadata": { }, "output_type": "execute_result" } ], "source": [ "preparse(\"i^2\")" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0" ] }, "execution_count": 7, "metadata": { }, "output_type": "execute_result" } ], "source": [ "e^(i*pi)+1" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "pi" ] }, "execution_count": 8, "metadata": { }, "output_type": "execute_result" } ], "source": [ "pi" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "%display latex" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\pi$ " ] }, "execution_count": 10, "metadata": { }, "output_type": "execute_result" } ], "source": [ "pi" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "## Calcul symbolique\n", "\n", "### Sous-section" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}x$ " ] }, "execution_count": 11, "metadata": { }, "output_type": "execute_result" } ], "source": [ "x" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sin\\left(x^{2}\\right)$ " ] }, "execution_count": 12, "metadata": { }, "output_type": "execute_result" } ], "source": [ "sin(x^2)" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}3.14159265358979$ " ] }, "execution_count": 13, "metadata": { }, "output_type": "execute_result" } ], "source": [ "n(pi)" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420199$ " ] }, "execution_count": 14, "metadata": { }, "output_type": "execute_result" } ], "source": [ "n(pi, digits=1000)" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420199\n" ] } ], "source": [ "print(n(pi, digits=1000))" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}e^{\\left(\\sqrt{163} \\pi\\right)}$ " ] }, "execution_count": 16, "metadata": { }, "output_type": "execute_result" } ], "source": [ "a = e^(pi*sqrt(163))\n", "a" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}2.62537412640768 \\times 10^{17}$ " ] }, "execution_count": 17, "metadata": { }, "output_type": "execute_result" } ], "source": [ "n(a)" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}2.6253741264076874399999999999925007259719818568889 \\times 10^{17}$ " ] }, "execution_count": 18, "metadata": { }, "output_type": "execute_result" } ], "source": [ "n(a, digits=50)" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sin\\left(4\\right)$ " ] }, "execution_count": 20, "metadata": { }, "output_type": "execute_result" } ], "source": [ "f(2)" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-0.756802495307928$ " ] }, "execution_count": 21, "metadata": { }, "output_type": "execute_result" } ], "source": [ "n(f(2))" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "f(x) = sin(x^2)\n", "f" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}x \\ {\\mapsto}\\ \\sin\\left(x^{2}\\right)$ " ] }, "execution_count": 19, "metadata": { }, "output_type": "execute_result" } ], "source": [ "f(x) = sin(x^2)\n", "f" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, x \\cos\\left(x^{2}\\right)$ " ] }, "execution_count": 22, "metadata": { }, "output_type": "execute_result" } ], "source": [ "diff(f(x), x)" ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}x \\ {\\mapsto}\\ 2 \\, x \\cos\\left(x^{2}\\right)$ " ] }, "execution_count": 23, "metadata": { }, "output_type": "execute_result" } ], "source": [ "diff(f, x)" ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "g = diff(f, x)" ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}x \\ {\\mapsto}\\ 2 \\, x \\cos\\left(x^{2}\\right)$ " ] }, "execution_count": 26, "metadata": { }, "output_type": "execute_result" } ], "source": [ "g" ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}6 \\, \\cos\\left(9\\right)$ " ] }, "execution_count": 27, "metadata": { }, "output_type": "execute_result" } ], "source": [ "g(3)" ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|<type|\\phantom{\\verb!x!}\\verb|'sage.symbolic.expression.Expression'>|$ " ] }, "execution_count": 28, "metadata": { }, "output_type": "execute_result" } ], "source": [ "type(x)" ] }, { "cell_type": "code", "execution_count": 32, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "u = var('u')" ] }, { "cell_type": "code", "execution_count": 35, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, u \\cos\\left(u^{2}\\right)$ " ] }, "execution_count": 35, "metadata": { }, "output_type": "execute_result" } ], "source": [ "g(u)" ] }, { "cell_type": "code", "execution_count": 36, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-4 \\, x^{2} \\sin\\left(x^{2}\\right) + 2 \\, \\cos\\left(x^{2}\\right)$ " ] }, "execution_count": 36, "metadata": { }, "output_type": "execute_result" } ], "source": [ "diff(f(x), x, 2)" ] }, { "cell_type": "code", "execution_count": 37, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "h(x,u) = u^3*exp(x)" ] }, { "cell_type": "code", "execution_count": 38, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( x, u \\right) \\ {\\mapsto} \\ u^{3} e^{x}$ " ] }, "execution_count": 38, "metadata": { }, "output_type": "execute_result" } ], "source": [ "h" ] }, { "cell_type": "code", "execution_count": 39, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( x, u \\right) \\ {\\mapsto} \\ 3 \\, u^{2} e^{x}$ " ] }, "execution_count": 39, "metadata": { }, "output_type": "execute_result" } ], "source": [ "diff(h, u)" ] }, { "cell_type": "code", "execution_count": 42, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{16} \\, \\sqrt{\\pi} {\\left(\\left(i + 1\\right) \\, \\sqrt{2} \\text{erf}\\left(\\left(\\frac{1}{2} i + \\frac{1}{2}\\right) \\, \\sqrt{2} \\pi\\right) + \\left(i - 1\\right) \\, \\sqrt{2} \\text{erf}\\left(\\left(\\frac{1}{2} i - \\frac{1}{2}\\right) \\, \\sqrt{2} \\pi\\right) - \\left(i - 1\\right) \\, \\sqrt{2} \\text{erf}\\left(\\sqrt{-i} \\pi\\right) + \\left(i + 1\\right) \\, \\sqrt{2} \\text{erf}\\left(\\left(-1\\right)^{\\frac{1}{4}} \\pi\\right)\\right)}$ " ] }, "execution_count": 42, "metadata": { }, "output_type": "execute_result" } ], "source": [ "integrate(f(x), (x, 0, pi))" ] }, { "cell_type": "code", "execution_count": 43, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}0.772651712690066$ " ] }, "execution_count": 43, "metadata": { }, "output_type": "execute_result" } ], "source": [ "n(_)" ] }, { "cell_type": "code", "execution_count": 41, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}3 \\, \\pi$ " ] }, "execution_count": 41, "metadata": { }, "output_type": "execute_result" } ], "source": [ "integrate(cos(x)^4, (x, 0, 8*pi))" ] }, { "cell_type": "code", "execution_count": 44, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\int_{0}^{+\\infty} e^{\\left(-x^{2}\\right)}\\,{d x}$ " ] }, "execution_count": 44, "metadata": { }, "output_type": "execute_result" } ], "source": [ "integrate(e^(-x^2), (x, 0, +oo), hold=True)" ] }, { "cell_type": "code", "execution_count": 45, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{2} \\, \\sqrt{\\pi}$ " ] }, "execution_count": 45, "metadata": { }, "output_type": "execute_result" } ], "source": [ "integrate(e^(-x^2), (x, 0, +oo))" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ ] } ], "metadata": { "kernelspec": { "display_name": "SageMath 7.6", "name": "sage-7.6" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.13" } }, "nbformat": 4, "nbformat_minor": 0 }