{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "[^ gor: Uvod](00_uvod.ipynb)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Kompleksna števila" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "V *Pythonu* so kompleksna števila vgrajena. Imaginarna enota se imenuje `j` in jo postavimo neposredno za številko." ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "(-1+0j)" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "1j**2 # 1j je v Pythonu imaginarna enota" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "complex" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "type(2+2.2j) #kompleksna števila imajo v Pythonu svoj tip" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Primer\n", "\n", "Zapiši število $$z=\\frac{1-2i}{1+i}$$ v obliki $x+iy$ in izračunaj $|z|$." ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "(-0.5-1.5j)" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "z = (1-2j)/(1+1j) #Python zlahka opravi z deljenjem kompleksnih števil, a računa približno s števili s plavajočo vejico\n", "z" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "1.5811388300841898" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "abs(z) # rezultat je podan v decimalnem zapisu" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Če želimo, da *Python* pusti rezultat v obliki korenov in ulomkov, potrebujemo knjižnico [sympy](http://www.sympy.org)." ] }, { "cell_type": "code", "execution_count": 42, "metadata": { "collapsed": true }, "outputs": [], "source": [ "import sympy as sym\n", "from sympy import I #uvozimo imaginarno enoto\n", "sym.init_printing() # lepši izpis formul" ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAADQAAAAQBAMAAACra0H4AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAzRAiu5mrdu/dZlSJ\nRDLkM64aAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAuUlEQVQYGWNgYBBiQAXMoS4PGBjYHBgYTD6h\nyjCoMfD8BUotYFAJg0kxGUDUJDEwdEBY7DAp7gKIQCcDQ/4DMBNDSl4BpxRQw3wFnXAghaGLgYH3\nD8OBG9ilOB2YNwRilwphYGL4DpfiERSUbBQUDAAKMLAnMDBwg9yNaddWBmYDxgQFLFJ8AQzsBvob\nDmCRsjxzNJPhPvMCBtaMn1lAY4EAFhry////ZGAphwhCSJgUshiUzbsAVRAA/hMwPFGxhOUAAAAA\nSUVORK5CYII=\n", "text/latex": [ "$$1 + 2 i$$" ], "text/plain": [ "1 + 2⋅ⅈ" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a = 1+2*I\n", "a" ] }, { "cell_type": "code", "execution_count": 32, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "(sympy.core.numbers.ImaginaryUnit, -1)" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(type(I),I**2)" ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAADYAAAAsBAMAAADcJlYTAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAzRAiu5mrdu/dZjKJ\nRFRer8KoAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABIUlEQVQ4EWNgYGBUYEAFpiHeQIEUIGb2V0CV\nYkpgOH+BgSGLgYHJNR9Njk2BgaMBqrweTY5jAgPbbxxyjJ9wywG1cH5iOloA0opuJlDo/gEe3gYc\nclMY7FgFEHKKM0FAGiTAwOrAoMCPw8wyoPx7BZAqDPv4AhiMGcSYsMptZWAoZvrHg02Oec6ZsAm8\nH6yAcmH90xaAlMAB4////ycwuIPdAhcclgygR3GBwedfIXQnsTnAREw+wVgwmm0BlKUSBpNjMoBJ\nwml2mBwfZmSTJKcTjtvMAzdwyjE/CMQpx8TwHU2OR1BQslFQMAAozAdzOAMDhjsZExRgGjHk9B8c\nwCl3n3kBVI414yewMAEBWLiwlEP4yCRMDlkMxuZdAGOBaAB9yWJNec1FwwAAAABJRU5ErkJggg==\n", "text/latex": [ "$$\\frac{1 - 2 i}{1 + i}$$" ], "text/plain": [ "1 - 2⋅ⅈ\n", "───────\n", " 1 + ⅈ " ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "z = (1-2*I)/(1+I)\n", "z" ] }, { "cell_type": "code", "execution_count": 29, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAACcAAAAvBAMAAACS3s5rAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAInarRM2ZVBDdiWbv\nuzJCz3LGAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABO0lEQVQoFcXSP0vDQBjH8W+MiRctUhc3qVJw\nsYODmw59B+3YwaFQRBDETq4pbg5C3UQQiuDiorMI+gqK4C6dXBUUiQX/PJcmlzbprM9w93s+XO4O\nLhBV4cdUTLCWRJOsqolJcJtJNunQpKGwLbkGame5Z1S1YSEAp6mODU5fc9QQvIULg3rLCcFH8Fux\nrkT4BhV9j8myDLKlXqk+BBcll6rS6yBo9eGlA9bBO3hiw4j6gpsIlawMP2cvz0mEyJ6+PohCfaYT\n4xnctXRjB14463uWYEsbs/39cNboNVU3bNis6dnd+FxHnRd7A/Trg3lkdPIj7b835mdJwuvfX2p+\n9ylzqFrlqpxWu0XuPo25NvZ3Gp1gDMoiTx4uU5VOhkB+hEy5DxmC4hibqjOX4WdYSqN1etlop9GR\n1zD4C/CPZPuhIDrOAAAAAElFTkSuQmCC\n", "text/latex": [ "$$\\frac{\\sqrt{10}}{2}$$" ], "text/plain": [ " ____\n", "╲╱ 10 \n", "──────\n", " 2 " ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "abs(z)" ] }, { "cell_type": "code", "execution_count": 34, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "sympy.core.add.Add" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from sympy import S,I\n", "z = 1+I\n", "type(z)" ] }, { "cell_type": "code", "execution_count": 35, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAABsAAAAVBAMAAABF8IgWAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAInarRM2ZVBDdiWbv\nuzJCz3LGAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAtUlEQVQYGWNggADG/yDwAcpjYHaAscC0KAqP\nIR2FyzEBhctWgMLtZGAQy70HF1rNwGHAsN4BygdqZW5g4DkA5TJtYOABivwDcl2AmJOBgfUHmMt0\nFMhVAqvi/MHAIKHvwMAAsdV/AVC0/wIDN4hmYJgLItg+MnA2gBkXQCTjN4ZdIJpBGUwy2DfMADHY\nExgEQfT75Qkg6hkDgyKIZj3aACQZZ61KBlvA/g0sCAwbiH2BIC4EAADhGSnmWnHssgAAAABJRU5E\nrkJggg==\n", "text/latex": [ "$$\\sqrt{2}$$" ], "text/plain": [ " ___\n", "╲╱ 2 " ] }, "execution_count": 35, "metadata": {}, "output_type": "execute_result" } ], "source": [ "abs(z)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Enačbe s kompleksnimi števili\n", "Reši enačbo\n", "$$iz^2-\\bar{z}=0$$\n", "\n", "Uporabimo ukaz `solve` iz knjižnice **sympy**." ] }, { "cell_type": "code", "execution_count": 50, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAABIAAAAUBAMAAAByuXB5AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAdt3NMolEEJlUImbv\nu6sslhSsAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAdElEQVQIHWMQMmEAgc1qDGEMDBWBLQxAGsiy\nZChaAGZxH2BgNQCzeB8wcH4Hs/gfMLD9BLPqGxjYPoJZ6wMYWL6iseoDYLL8DQycEB1MFxi4IaYw\nHmDggJjMoMlQ4wDWy1Aq+ATqArCzwG7BwxJSAUsKqQEA5KQjcutb2KEAAAAASUVORK5CYII=\n", "text/latex": [ "$$\\left [ 0\\right ]$$" ], "text/plain": [ "[0]" ] }, "execution_count": 50, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from sympy import Symbol,Eq,conjugate,solve,im,re\n", "z = Symbol('z')\n", "solve(Eq(I*z**2-conjugate(z),0),z)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "Python ni znal poiskati vseh rešitev, zato poskusimo kompleksno enačbo najprej preoblikovati v dve realni enačbi." ] }, { "cell_type": "code", "execution_count": 48, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPEAAAAcBAMAAABCP0RmAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRK5CYII=\n", "text/latex": [ "$$- 2 x y - x + i \\left(x^{2} - y^{2} + y\\right)$$" ], "text/plain": [ " ⎛ 2 2 ⎞\n", "-2⋅x⋅y - x + ⅈ⋅⎝x - y + y⎠" ] }, "execution_count": 48, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x,y = sym.symbols(\"x,y\",real=True) # deklariramo realne spremenljivke\n", "z = x+y*I\n", "lhs = sym.expand_complex(I*z**2-conjugate(z))\n", "lhs" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Posebej izenačimo realni del in imaginarni del enačbe." ] }, { "cell_type": "code", "execution_count": 56, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAATEAAAAbBAMAAADmC6W/AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nTkSuQmCC\n", "text/latex": [ "$$\\left [ - 2 x y - x = 0, \\quad x^{2} - y^{2} + y = 0\\right ]$$" ], "text/plain": [ "⎡ 2 2 ⎤\n", "⎣-2⋅x⋅y - x = 0, x - y + y = 0⎦" ] }, "execution_count": 56, "metadata": {}, "output_type": "execute_result" } ], "source": [ "enacbe = [Eq(re(lhs),0),Eq(im(lhs),0)]\n", "enacbe" ] }, { "cell_type": "code", "execution_count": 57, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAgcAAAA/BAMAAACGOXbhAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", "text/latex": [ "$$\\left [ \\left ( 0, \\quad 0\\right ), \\quad \\left ( 0, \\quad 1\\right ), \\quad \\left ( - \\frac{\\sqrt{3}}{2}, \\quad - \\frac{1}{2}\\right ), \\quad \\left ( \\frac{\\sqrt{3}}{2}, \\quad - \\frac{1}{2}\\right )\\right ]$$" ], "text/plain": [ "⎡ ⎛ ___ ⎞ ⎛ ___ ⎞⎤\n", "⎢ ⎜-╲╱ 3 ⎟ ⎜╲╱ 3 ⎟⎥\n", "⎢(0, 0), (0, 1), ⎜───────, -1/2⎟, ⎜─────, -1/2⎟⎥\n", "⎣ ⎝ 2 ⎠ ⎝ 2 ⎠⎦" ] }, "execution_count": 57, "metadata": {}, "output_type": "execute_result" } ], "source": [ "resitve = solve(enacbe,[x,y])\n", "resitve # pari (x,y), ki rešijo sistem enačb" ] }, { "cell_type": "code", "execution_count": 58, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAR0AAAA/BAMAAAAvVwHBAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", "text/latex": [ "$$\\left [ 0, \\quad i, \\quad - \\frac{\\sqrt{3}}{2} - \\frac{i}{2}, \\quad \\frac{\\sqrt{3}}{2} - \\frac{i}{2}\\right ]$$" ], "text/plain": [ "⎡ ___ ___ ⎤\n", "⎢ ╲╱ 3 ⅈ ╲╱ 3 ⅈ⎥\n", "⎢0, ⅈ, - ───── - ─, ───── - ─⎥\n", "⎣ 2 2 2 2⎦" ] }, "execution_count": 58, "metadata": {}, "output_type": "execute_result" } ], "source": [ "[r[0]+I*r[1] for r in resitve] # rešitve prvotne enačbe kot kompleksna števila" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Množice v kompleksni ravnini\n", "\n", "Nobena knjižnica v *Python-u* ne zna risati neposredno v kompleksni ravnini. Zato uporabimo nastavek $z=x+iy$ in vse skupaj prevedemo v ravnino $x,y$.\n", "\n", "### Primer\n", "Poišči vse kompleksne rešitve spodnje enačbe, tj. opiši ali skiciraj množico\n", "rešitev v $\\mathbb{C}$.\n", "\n", "$$ Im\\left(\\frac{1}{z}\\right)=1$$" ] }, { "cell_type": "code", "execution_count": 64, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAADcAAAASBAMAAAANlFvwAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\n75nQ6/gxAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA7klEQVQoFWNgwAKYN2ARhAkxF8BYSLQIEhuD\nuQhDBEkAU1JI2VVNAaICKin2iIExT4HBV4CBMYG9iWMCimRCLAMb1wUGfQYGNgG2j1wOyJIsC14y\nePFNYNjNwMDIwAn3FcRYRoZPDAK8CkASCPgVwPoyZ87snTlzHpDN9ZGBIV6A5w9I+PwBBgGwNAPU\nQZwGAgwvGZguAJUJ6DMwoUrKL0hg2MzAbQDUv2A/gztEI0xnPEsBw2uG+AIGBmEloXQgBQZQY7mV\ngMJJ7x2golAKKgnhvUSVYwiH85knMH6Fc9AZHArcCuhicD6r0hsgGwAj4y3aHKrxQgAAAABJRU5E\nrkJggg==\n", "text/latex": [ "$$x + i y$$" ], "text/plain": [ "x + ⅈ⋅y" ] }, "execution_count": 64, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x,y = sym.symbols(\"x,y\",real=True)\n", "z = x+I*y\n", "z" ] }, { "cell_type": "code", "execution_count": 65, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHsAAAApBAMAAADnr0uwAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nI30AAAAASUVORK5CYII=\n", "text/latex": [ "$$- \\frac{y}{x^{2} + y^{2}} = 1$$" ], "text/plain": [ " -y \n", "─────── = 1\n", " 2 2 \n", "x + y " ] }, "execution_count": 65, "metadata": {}, "output_type": "execute_result" } ], "source": [ "enacba = Eq(im(1/z),1)\n", "enacba" ] }, { "cell_type": "code", "execution_count": 68, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHkAAAApBAMAAADjWpuNAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nkXc/A5n9f1XKS/9bAAAAAElFTkSuQmCC\n", "text/latex": [ "$$- \\frac{y}{x^{2} + y^{2}} - 1$$" ], "text/plain": [ " y \n", "- ─────── - 1\n", " 2 2 \n", " x + y " ] }, "execution_count": 68, "metadata": {}, "output_type": "execute_result" } ], "source": [ "leva = enacba.lhs-enacba.rhs # vse damo na eno stran\n", "leva" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Množica rešitev je podana implicitno z enačbo. Zato uporabimo `contour` iz knjižnice [matplotlib](http://www.matplotlib.org)." ] }, { "cell_type": "code", "execution_count": 82, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 82, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "import numpy as np\n", "%matplotlib inline\n", "# generiramo mrežo točk (x,y) v kateri bomo izračunali vrednosti leve strani enačbe\n", "xi = np.linspace(-1,1)\n", "X,Y = np.meshgrid(xi,xi) # pripravimo tabele, ki jih lahko direktno vstavimo v funkcijo\n", "fun = sym.lambdify((x,y),leva) # izraz spremenimo v Python funkcijo\n", "plt.contour(X,Y,fun(X,Y),levels=[0]) # narišemo množico točk, pri katerih je leva stran enaka 0 (levels=[0])\n", "plt.axis(\"equal\")\n", "plt.title(\"množica je krožnica\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Primer\n", "Skiciraj množico rešitev neenačbe \n", " $$ |z-3i+4|\\le 4$$\n", "v kompleksni ravnini." ] }, { "cell_type": "code", "execution_count": 108, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQUAAAAaBAMAAABMT1RYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n99S8/b5en93o+2+J8wrOJ1icJLelBAAAAABJRU5ErkJggg==\n", "text/latex": [ "$$\\sqrt{x^{2} + 8 x + y^{2} - 6 y + 25} - 4$$" ], "text/plain": [ " __________________________ \n", " ╱ 2 2 \n", "╲╱ x + 8⋅x + y - 6⋅y + 25 - 4" ] }, "execution_count": 108, "metadata": {}, "output_type": "execute_result" } ], "source": [ "leva = abs(z-3*I+4)-4\n", "leva" ] }, { "cell_type": "code", "execution_count": 113, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 113, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# generiramo mrežo točk (x,y) v kateri bomo izračunali vrednosti leve strani enačbe\n", "xi = np.linspace(-8,0)\n", "yi = np.linspace(-2,8)\n", "X,Y = np.meshgrid(xi,yi) # pripravimo tabele, ki jih lahko direktno vstavimo v funkcijo\n", "fun = sym.lambdify((x,y),leva) # izraz spremenimo v Python funkcijo\n", "fun = np.vectorize(fun) # Poskrbimo, da funkcija sprejme tudi tabele\n", "plt.contourf(X,Y,fun(X,Y),levels=[0,-4]) # narišemo množico točk, pri katerih je leva stran med 0 in -4 (levels=[0,-4])\n", "plt.axis(\"equal\")\n", "plt.title(\"množica je poln krog\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Faktorizacija polinoma\n", "Reši enačbo \n", "$$z^4 + 4 = 0,$$ nato pa razstavi polinom $z^4 + 4$ na dva kvadratna\n", "faktorja z realnimi koeficienti." ] }, { "cell_type": "code", "execution_count": 115, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAS0AAAAUBAMAAAAqz3YMAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nit8AAAAASUVORK5CYII=\n", "text/latex": [ "$$\\left [ -1 - i, \\quad -1 + i, \\quad 1 - i, \\quad 1 + i\\right ]$$" ], "text/plain": [ "[-1 - ⅈ, -1 + ⅈ, 1 - ⅈ, 1 + ⅈ]" ] }, "execution_count": 115, "metadata": {}, "output_type": "execute_result" } ], "source": [ "z = Symbol('z')\n", "resitve = solve(z**4+4,z)\n", "resitve" ] }, { "cell_type": "code", "execution_count": 125, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYIAAAAVBAMAAAC5wPT1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nyQzAUak7NQD4Gcs2sKd/ZYUMOcGp3JEAAAAASUVORK5CYII=\n", "text/latex": [ "$$\\left(z - 1 - i\\right) \\left(z - 1 + i\\right) \\left(z + 1 - i\\right) \\left(z + 1 + i\\right)$$" ], "text/plain": [ "(z - 1 - ⅈ)⋅(z - 1 + ⅈ)⋅(z + 1 - ⅈ)⋅(z + 1 + ⅈ)" ] }, "execution_count": 125, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sym.prod([z-z0 for z0 in resitve])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Polinom $z^4+4$ lahko zapišemo kot produkt linearnih faktorjev $(z-z_i)$, kjer so $z_i$ ničle polinoma. Rešitve nastopajo kot konjugirani pari $-1\\pm i$ in $1\\pm i$. Če skupaj zmnožimo linearne faktorje, ki pripadajo konjugiranemu paru, dobimo razcep na kvadratne faktorje z linearnimi koeficienti." ] }, { "cell_type": "code", "execution_count": 127, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAO8AAAAcBAMAAAB7WPU9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", "text/latex": [ "$$\\left(z^{2} - 2 z + 2\\right) \\left(z^{2} + 2 z + 2\\right)$$" ], "text/plain": [ "⎛ 2 ⎞ ⎛ 2 ⎞\n", "⎝z - 2⋅z + 2⎠⋅⎝z + 2⋅z + 2⎠" ] }, "execution_count": 127, "metadata": {}, "output_type": "execute_result" } ], "source": [ "izraz = sym.expand((z-resitve[0])*(z-resitve[1]))*sym.expand((z-resitve[2])*(z-resitve[3]))\n", "izraz" ] }, { "cell_type": "code", "execution_count": 128, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAADcAAAAWBAMAAACWBRnmAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMARN3vMmYQu3aZqyJU\nic384Ct+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA1UlEQVQoFWNggAB2KI2VOp+AVRgsyHYfjyRH\nPh5JH3ySF/BIsm7AlGSZAHViBgOmJFMBVPLkzPlTIUy2DVAhhmaYJAPD/gSIIE8DVJI7Di7JWr8G\nTZKXDyTJqCzgClUNpOA6o8GS6R0zF2BKsjWAJRMYdiHk4Dp5GcCSDCxIGuGSR6CSknCNvO/ePbJ7\n924CAwPbBYgkD5CNAFAH8axatd6qgYHhJAPDAgxJoAAz0Cssqlf7BLBJ8gMlORmCHyLk4A5i4K3/\nFYAkDmbCAwFdAsRnRFYOAGauMW9wQhasAAAAAElFTkSuQmCC\n", "text/latex": [ "$$z^{4} + 4$$" ], "text/plain": [ " 4 \n", "z + 4" ] }, "execution_count": 128, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# če kvadratna faktorja zmnožimo, dobimo prvotni polinom\n", "sym.expand(izraz)" ] }, { "cell_type": "code", "execution_count": 129, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAO8AAAAcBAMAAAB7WPU9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", "text/latex": [ "$$\\left(z^{2} - 2 z + 2\\right) \\left(z^{2} + 2 z + 2\\right)$$" ], "text/plain": [ "⎛ 2 ⎞ ⎛ 2 ⎞\n", "⎝z - 2⋅z + 2⎠⋅⎝z + 2⋅z + 2⎠" ] }, "execution_count": 129, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# razcep na kvadratne faktorje lahko izračunamo tudi s funkcijo factor\n", "sym.factor(z**4+4)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Polarni zapis\n", "Kompleksno število lahko zapišemo tudi v polarni obliki\n", "$$ z =r e^{i\\phi} = r(\\cos(\\phi) + i\\sin(\\phi)).$$\n", "Polarna oblika je primerna za množenje, potenciranje in korene.\n", "### Primer\n", "Poišči vse korene \n", "$$\\sqrt[3]{-27+27i}.$$\n", "Iščemo rešitve enačbe\n", "$$ z^3 = -27+27i.$$" ] }, { "cell_type": "code", "execution_count": 140, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAADQAAAAUBAMAAAAw+gPuAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMARImrEHa7zVTvMt2Z\nImbh7FZmAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABIUlEQVQoFWNgAANVZQcIgynIhIEhBsj2UYUI\n8DzgSYCwzjEcY2BYC2SrKkGlFPg+QFiHGN5DGIUQCkjyFMCYxyGMLzA+g+cCKJNlFoOoAQPDsz8P\noAKupTCZ8AAGFaAOPrgpDDBnMDAc4/XYALTgAFQtAwPXDxhzfQNbAAMD2wMo33MB7z8wE2jT/Qf+\nXgwMnAJQKe4LfFMgzL0MLx3amRgY/B2gUlyqyhcgzEVBtgzOugwM0kAuo7KjKtBsEHAKPQthAMk3\nQMzEVsE2ASzCuJfhEkyqzgrIWsFtwLgALHKzgeUUTApskIM/zC2WT+McYFIQuh/G/QxjwGlocDIw\nfIILwRiTkBisMDaY5v0I44YzsCAcDxJk3ACTYj36DsYE0wC23juM8GxjbgAAAABJRU5ErkJggg==\n", "text/latex": [ "$$r^{3} e^{3 i f}$$" ], "text/plain": [ " 3 3⋅ⅈ⋅f\n", "r ⋅ℯ " ] }, "execution_count": 140, "metadata": {}, "output_type": "execute_result" } ], "source": [ "r,f = sym.symbols(\"r,f\",real=True)\n", "z = r*sym.exp(I*f)\n", "z**3" ] }, { "cell_type": "code", "execution_count": 145, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAANYAAAAZBAMAAAC/c818AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nCtzKFfDZKF25gtYnd6yXRcE4C/Xp+AMChPZtJXbVSQAAAABJRU5ErkJggg==\n", "text/latex": [ "$$i r^{3} \\sin{\\left (3 f \\right )} + r^{3} \\cos{\\left (3 f \\right )}$$" ], "text/plain": [ " 3 3 \n", "ⅈ⋅r ⋅sin(3⋅f) + r ⋅cos(3⋅f)" ] }, "execution_count": 145, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sym.expand_complex(z**3)" ] }, { "cell_type": "code", "execution_count": 149, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAADEAAAAVBAMAAAAdjxsPAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABLklEQVQoFXWQsUoDQRRFz0YTk81GomChNtGA\nnRDQDwiWNtnKwiaLYGMTEWErNaWdtppmQcRS/QJjYyUY/AHtLI3BoNisd2aRQCQP5r679zAzbxaG\ny2nHprrDOWSa/7MkmR8FWB5F0tEokur8EWdxpcr6VhjuJsmm2nR5VbrBxDfvGjNIyCGkA46asA1n\nLGjYBKQjWZ/cKZxDo6qT9hPivUIuIvMJd74IZNtwLOhq5fuWyF37MAZeS/5RS+X2jWa/JLp45qkJ\nkbzqxE7qVjROV5/1CpM2gQvLy1LvRpL6wPVtlKqY5gUSsw+nx5ua6sHqM06RemB8w7+0UaHEFIyX\n8IrUSiaqHdjGGtzDXLi3Ay82yrfsNc5VuBTppXH8A7MdbaHQM0pefzGybiC3A8svYtdF97bi7PwA\nAAAASUVORK5CYII=\n", "text/latex": [ "$$27 \\sqrt{2}$$" ], "text/plain": [ " ___\n", "27⋅╲╱ 2 " ] }, "execution_count": 149, "metadata": {}, "output_type": "execute_result" } ], "source": [ "z0 = -27+27*I \n", "abs(z0)" ] }, { "cell_type": "code", "execution_count": 150, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAABcAAAArBAMAAABhvA5FAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIom7VJlmdt1E7xDN\nMqsI8sYEAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA00lEQVQoFWNgEFIyYYABxgAG/wQYh/0LA38D\njMO5kmH/ARgHSCOUATlHGRjq/////xXI5NUIYGDSdRcrFAAr1rpwhUGR/QKYzcC1moHBgG8DkMMo\nwMD8hYH3AQdIgv83A/M3BtYEfpAMlwEDx0eG/Rv4JwA5bAkM/gUMFQxcCkAOw9RQSwYGMwY+MAck\nQAcA9BUMfKCWdbwBSCZxGCBxqpA4jDlIHCYeJE4iEoe7AInDxIDE8UXicCsgcVjevXu/rgBhKzDE\nEYAficNk/y8BIQNhAQA+9DM97RJrAQAAAABJRU5ErkJggg==\n", "text/latex": [ "$$\\frac{3 \\pi}{4}$$" ], "text/plain": [ "3⋅π\n", "───\n", " 4 " ] }, "execution_count": 150, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sym.arg(z0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Enačbe dobimo tako, da izenačimo $r^3$ z $|z_0|$ in $3\\phi$ z $\\arg(z_0)$. V enačbi za argument, lahko eni strani dodamo večkratnik $2\\pi$:\n", "$$r^3 = 27\\sqrt{2}$$\n", "$$3\\phi = \\frac{3\\pi}{4}+2k\\pi$$\n", "\n", "Rešitev je neskončno\n", "$$r = 3\\sqrt[6]{2} $$\n", "$$\\phi = \\frac{\\pi}{4} +\\frac{2k\\pi}{3}$$" ] }, { "cell_type": "code", "execution_count": 167, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "-3.25265\t+\t0.87154i\n", "\n", "0.87154\t+\t-3.25265i\n", "\n", "2.38110\t+\t2.38110i\n", "\n", "-3.25265\t+\t0.87154i\n", "\n", "0.87154\t+\t-3.25265i\n", "\n", "2.38110\t+\t2.38110i\n", "\n", "-3.25265\t+\t0.87154i\n", "\n", "0.87154\t+\t-3.25265i\n", "\n", "2.38110\t+\t2.38110i\n", "\n", "-3.25265\t+\t0.87154i\n", "\n" ] } ], "source": [ "# Če tabeliramo različne rešitve za vrednosti k = -5,-4,...,0,1,..\n", "from sympy import exp\n", "for k in range(-5,5):\n", " zi = complex(abs(z0)**(1/3)*exp(1j*pi*(1/4+2*k/3)))\n", " print(\"%0.5f\\t+\\t%0.5fi\\n\" % (re(zi),im(zi)))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Čeprav je rešitev za $\\phi$ neskončno, je kompleksnih rešitev le končno mnogo - 3, saj se začnejo ponavljati ($e^{i\\phi}$ je periodična funkcija s periodo $2\\pi$)." ] }, { "cell_type": "code", "execution_count": 170, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAzIAAAA/BAMAAADJbeBdAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n/j+jDXoUEUMsRgAAAABJRU5ErkJggg==\n", "text/latex": [ "$$\\left [ \\frac{3}{2} 2^{\\frac{2}{3}} + \\frac{3 i}{2} 2^{\\frac{2}{3}}, \\quad - \\frac{3}{4} 2^{\\frac{2}{3}} + \\frac{3 \\sqrt{3}}{4} 2^{\\frac{2}{3}} - \\frac{3 i}{4} 2^{\\frac{2}{3}} \\sqrt{3} - \\frac{3 i}{4} 2^{\\frac{2}{3}}, \\quad - \\frac{3 \\sqrt{3}}{4} 2^{\\frac{2}{3}} - \\frac{3}{4} 2^{\\frac{2}{3}} - \\frac{3 i}{4} 2^{\\frac{2}{3}} + \\frac{3 i}{4} 2^{\\frac{2}{3}} \\sqrt{3}\\right ]$$" ], "text/plain": [ "⎡ 2/3 2/3 2/3 2/3 ___ 2/3 ___ 2/3 \n", "⎢3⋅2 3⋅2 ⋅ⅈ 3⋅2 3⋅2 ⋅╲╱ 3 3⋅2 ⋅╲╱ 3 ⋅ⅈ 3⋅2 ⋅ⅈ 3⋅2\n", "⎢────── + ────────, - ────── + ──────────── - ────────────── - ────────, - ───\n", "⎣ 2 2 4 4 4 4 \n", "\n", "2/3 ___ 2/3 2/3 2/3 ___ ⎤\n", " ⋅╲╱ 3 3⋅2 3⋅2 ⋅ⅈ 3⋅2 ⋅╲╱ 3 ⋅ⅈ⎥\n", "───────── - ────── - ──────── + ──────────────⎥\n", " 4 4 4 4 ⎦" ] }, "execution_count": 170, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# rešitve seveda lahko piščemo tudi s solve\n", "z = Symbol(\"z\")\n", "solve(Eq(z**3,-27+27*I),z)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "[<< nazaj: neenačbe](01b_neenacbe.ipynb)" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\n", "
\n", "\n", "\n", "\n", "\n", "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import disqus\n", "%reload_ext disqus\n", "%disqus matpy" ] } ], "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": "3.5.0+" } }, "nbformat": 4, "nbformat_minor": 0 }