{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Nombres complexes\n", "\n", "Ce notebook est relatif au premier chapitre de cours de Prolégomènes de Mathématiques du [Master Sciences et Numérique pour la Santé](https://sns.edu.umontpellier.fr).\n", "\n", "> **⚠️ Rappel ⚠️**: Ce chapitre a pour objectif de synthétiser les notions et formules essentielles à l’utilisation des nombres complexes. Dans le cadre du Master, elles trouvent leurs applications dans les\n", "modules suivants : HAH807E - Chaîne d’Acquisition / Traitement du Signal, HAH710E - Ingénierie\n", ": Électronique, HAH802E - Capteurs et Traitement d’Images, HAH808E - Capteurs US\n", "pour le Biomédical, HAH805E - Robotique médicale, HAH806E - Neuroprothèse, HAH906E -\n", "Acquisition de données et traitement du signal pour le biomédical, HAH908E - Thermographie\n", "IR, détecteurs optiques et US, HAH909E - Robotique médicale 2, HAH910E - Neuroprothèse 2.\n", "\n", "Notebook réalisé par E. Le Clézio, le 21 août 2023. \n", "[Master Sciences et Numérique pour la Santé](https://sns.edu.umontpellier.fr) \n", "[Université de Montpellier](https://www.umontpellier.fr)\n", "\n", "## Import des bibliothèques nécessaires au fonctionnement du Notebook" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exercice 1 :\n", "\n", "Soient les nombres complexes $z_1=2-i$ et $z_2=1+2i$.\n", "1. Quelles sont les parties réelles et imaginaires de $z_1$ et $z_2$ ?\n", "2. Calculer $z_1+z_2$, $z_1-z_2$, $z_1*z_2$ et $\\frac{z_1}{z_2}$.\n", "\n" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1.\n", "La partie réelle de z1 est : 2.0\n", "La partie imaginaire de z1 est : -1.0\n", "La partie réelle de z2 est : 1.0\n", "La partie imaginaire de z2 est : 2.0\n", "2.\n", "z1+z2 = (3+1j)\n", "z1-z2 = (1-3j)\n", "z1*z2 = (4+3j)\n", "z1/z2 = -1j\n" ] } ], "source": [ "z1 = 2-1j\n", "z2 = 1+2j\n", "\n", "print('1.')\n", "print('La partie réelle de z1 est :', np.real(z1))\n", "print('La partie imaginaire de z1 est :', np.imag(z1))\n", "print('La partie réelle de z2 est :', np.real(z2))\n", "print('La partie imaginaire de z2 est :', np.imag(z2))\n", "\n", "print('2.')\n", "\n", "print('z1+z2 = ', z1+z2)\n", "print('z1-z2 = ', z1-z2)\n", "print('z1*z2 = ', z1*z2)\n", "print('z1/z2 = ', z1/z2)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exercice 2 :\n", "\n", "Soient les nombres complexes $z_1=2-i$ et $z_2=1+2i$.\n", "1. Quels sont les conjugués $\\overline{z_1}$ et $\\overline{z_2}$ de $z_1$ et $z_2$ ?\n", "2. Vérifier que les relations ci-dessus sont bien vérifiées.\n" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1.\n", "Le conjugé de z1 est : (2+1j)\n", "Le conjugé de z2 est : (1-2j)\n", "2.\n" ] } ], "source": [ "print('1.')\n", "print('Le conjugé de z1 est :', np.conj(z1))\n", "print('Le conjugé de z2 est :', np.conj(z2))\n", "print('2.')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\\begin{eqnarray}\n", "\t\\overline{z+z'}&=&\\overline{z}+\\overline{z'}, \\\\\n", "\t\\overline{-z}&=&-\\overline{z}, \\\\\n", "\t\\overline{zz'}&=&\\overline{z}\\overline{z'}, \\\\\n", "\t\\overline{z^n}&=&\\overline{z}^n, \\\\\n", "\\overline{\\left(\\frac{z}{z'}\\right)} &=& \\frac{\\overline{z}}{\\overline{z'}}.\n", "\\end{eqnarray}\n", "$$" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0j\n", "0j\n", "0j\n", "0j\n", "0j\n" ] } ], "source": [ "print(np.conj(z1+z2)-(np.conj(z1)+np.conj(z2)))\n", "print(np.conj(-z1)-(-np.conj(z1)))\n", "print(np.conj(z1*z2)-(np.conj(z1)*np.conj(z2)))\n", "print(np.conj(z1**10)-(np.conj(z1)**10))\n", "print(np.conj(z1/z2)-(np.conj(z1)/np.conj(z2)))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exercice 3 :\n", "\n", "Soit le nombre complexe $z=a+ib$. Calculer $z\\overline{z}$.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$(a+ib)*(a-ib) = a^2-aib + iba -(ib)^2 \\\\\n", "= a^2-iab +iab -i^2b^2 \\\\\n", "= a^2 + b^2. $$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exercice 4 :\n", "\n", "La définition précédente aurait-elle pu être prévue par un raisonnement géométrique ?\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Il s'agit en fait du théorème de Pythagore. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exercice 5 :\n", "\n", "Soient les nombres complexes $z_1=2-i$ et $z_2=1+2i$. Calculer leurs modules et arguments." ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Le module au carré de z1 est : 5.000000000000001\n", "Le module de z1 est : 2.23606797749979\n", "L'argument de z1 est : -0.4636476090008061 [rad], soient -26.56505117707799 °.\n", "Le module de au carré z2 est : 5.000000000000001\n", "Le module de z2 est : 2.23606797749979\n", "L'argument de z2 est : 1.1071487177940904 [rad], soient 63.43494882292201 °.\n", "3.141592653589793\n" ] } ], "source": [ "print('Le module au carré de z1 est :', np.abs(z1)**2)\n", "print('Le module de z1 est :', np.abs(z1))\n", "print(\"L'argument de z1 est :\", np.angle(z1), '[rad], soient ', np.angle(z1)*180/np.pi, '°.')\n", "print('Le module de au carré z2 est :', np.abs(z2)**2)\n", "print('Le module de z2 est :', np.abs(z2))\n", "print(\"L'argument de z2 est :\", np.angle(z2), '[rad], soient ', np.angle(z2)*180/np.pi, '°.')\n", "print((-np.angle(z1)+np.angle(z2))*2)" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": "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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "figure, axes = plt.subplots(figsize = (8,8))\n", "Drawing_uncolored_circle = plt.Circle( (0, 0), 1, fill = False )\n", " \n", "axes.set_aspect( 1 )\n", "axes.add_artist( Drawing_uncolored_circle )\n", "#plt.plot([2, -1], [1, 2], 'k--')\n", "#plt.plot([0, 0], [-5, 5], 'k--')\n", "\n", "plt.arrow(0, 0, 2, -1, head_width=0.05, head_length=0.1, fc='k', ec='k')\n", "plt.arrow(0, 0, 1, 2, head_width=0.05, head_length=0.1, fc='k', ec='k')\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exercice 6 :\n", "\n", "Déterminer, dans le plan complexe, les positions des deux racines carrées d'un nombre complexe $z$. Vous vous aiderez du cercle trigonométrique pour cela." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$ z = |z|e^{i\\phi+2i\\pi} \\\\\n", "z^{1/2} = |z|^{1/2}(e^{i\\phi+2ik\\pi)})^{1/2} \\\\\n", "z^{1/2} = \\sqrt{|z|}(e^{(i\\phi+2ik\\pi))/2}) \\\\\n", "z^{1/2} = \\sqrt{|z|}e^{(i\\phi/2+ik\\pi))} \\\\ \n", "z^{1/2} = \\{ \\sqrt{|z|}e^{(i\\phi/2+i0\\pi))} ; \\sqrt{|z|}e^{(i\\phi/2+i1\\pi))} \\} \\\\\n", "z^{1/2} = \\{ \\sqrt{|z|}e^{(i\\phi/2))} ; \\sqrt{|z|}e^{(i\\phi/2+i\\pi))} \\} \\\\\n", "z^{1/2} = \\{ \\sqrt{|z|}e^{(i\\phi/2))} ; -\\sqrt{|z|}e^{(i\\phi/2))} \\}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exercice 7 :\n", "\n", "Déterminer, dans le plan complexe, les positions des racines $n^{\\mathrm{\\grave{e}me}}$ d'un nombre complexe $z$. Vous vous aiderez du cercle trigonométrique pour cela." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$z = |z|e^{i\\phi+2i\\pi} \\\\\n", "z^{1/n} = |z|^{1/n}(e^{i\\phi+2ik\\pi)})^{1/n} \\\\\n", "z^{1/n} = \\sqrt[n]{|z|}(e^{(i\\phi+2ik\\pi))/n}) \\\\\n", "z^{1/n} = \\sqrt[n]{|z|}e^{(i\\phi/n+i2k/n\\pi))} \\\\ \n", "z^{1/n} = \\{ \\sqrt[n]{|z|}e^{(i\\phi/n+i0\\pi))} ; ... ; \\sqrt[n]{|z|}e^{(i\\phi/n+i(n-1)/n\\pi))} \\} \\\\\n", "z^{1/n} = \\{ \\sqrt[n]{|z|}e^{(i\\phi/n))} ; ... ; \\sqrt[n]{|z|}e^{(i\\phi/n+i(n-1)/n\\pi))} \\}. $$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Application : racines cubiques de 2" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "figure, axes = plt.subplots(figsize = (8,8))\n", "Drawing_uncolored_circle = plt.Circle( (0, 0), 1, fill = False )\n", " \n", "axes.set_aspect( 1 )\n", "axes.add_artist( Drawing_uncolored_circle )\n", "plt.plot([-5, 5], [0, 0], 'k--')\n", "plt.plot([0, 0], [-5, 5], 'k--')\n", "\n", "plt.arrow(0, 0, 0, 0.5, head_width=0.05, head_length=0.1, fc='k', ec='k')\n", "plt.arrow(0, 0, 0.5, 0, head_width=0.05, head_length=0.1, fc='k', ec='k')\n", "\n", "plt.plot(2, 0, 'ko')\n", "plt.plot(np.power(2, 1/3)*np.cos(0), np.power(2, 1/3)*np.sin(0), 'rx')\n", "plt.plot(np.power(2, 1/3)*np.cos(2*np.pi/3), np.power(2, 1/3)*np.sin(2*np.pi/3), 'rx')\n", "plt.plot(np.power(2, 1/3)*np.cos(4*np.pi/3), np.power(2, 1/3)*np.sin(4*np.pi/3), 'rx')\n", "\n", "plt.arrow(0, 0, (np.power(2, 1/3)-0.1)*np.cos(0), (np.power(2, 1/3)-0.1)*np.sin(0), head_width=0.05, head_length=0.1, fc='r', ec='r')\n", "plt.arrow(0, 0, (np.power(2, 1/3)-0.1)*np.cos(2*np.pi/3), (np.power(2, 1/3)-0.1)*np.sin(2*np.pi/3), head_width=0.05, head_length=0.1, fc='r', ec='r')\n", "plt.arrow(0, 0, (np.power(2, 1/3)-0.1)*np.cos(4*np.pi/3), (np.power(2, 1/3)-0.1)*np.sin(4*np.pi/3), head_width=0.05, head_length=0.1, fc='r', ec='r')\n", "\n", "Drawing_uncolored_circle = plt.Circle( (0, 0), np.power(2, 1/3), fill = False, linestyle = '--')\n", "axes.add_artist( Drawing_uncolored_circle )\n", "\n", "plt.xlim([-2.5,2.5])\n", "plt.ylim([-2.5,2.5])\n", "plt.xlabel('Re(z)')\n", "plt.ylabel('Im(z)')\n", "plt.title( 'Racines cubiques de 2' )\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Application : racines cinquième de 2 + i" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "figure, axes = plt.subplots(figsize = (8,8))\n", "Drawing_uncolored_circle = plt.Circle( (0, 0), 1, fill = False )\n", " \n", "axes.set_aspect( 1 )\n", "axes.add_artist( Drawing_uncolored_circle )\n", "plt.plot([-5, 5], [0, 0], 'k--')\n", "plt.plot([0, 0], [-5, 5], 'k--')\n", "\n", "plt.arrow(0, 0, 0, 0.5, head_width=0.05, head_length=0.1, fc='k', ec='k')\n", "plt.arrow(0, 0, 0.5, 0, head_width=0.05, head_length=0.1, fc='k', ec='k')\n", "\n", "plt.plot(2, 1, 'ko')\n", "plt.arrow(0, 0, 2, 1, head_width=0, head_length=0, fc='k', ec='k', linestyle = '--')\n", "\n", "plt.plot(np.power(np.abs(2+1j), 1/5)*np.real(np.exp(1j*np.angle(2+1j))), np.power(np.abs(2+1j), 1/5)*np.imag(np.exp(1j*np.angle(2+1j))), 'rx')\n", "plt.plot(np.power(np.abs(2+1j), 1/5)*np.real(np.exp(1j*(np.angle(2+1j)+2*np.pi/5))), np.power(np.abs(2+1j), 1/5)*np.imag(np.exp(1j*(np.angle(2+1j)+2*np.pi/5))), 'rx')\n", "plt.plot(np.power(np.abs(2+1j), 1/5)*np.real(np.exp(1j*(np.angle(2+1j)+4*np.pi/5))), np.power(np.abs(2+1j), 1/5)*np.imag(np.exp(1j*(np.angle(2+1j)+4*np.pi/5))), 'rx')\n", "plt.plot(np.power(np.abs(2+1j), 1/5)*np.real(np.exp(1j*(np.angle(2+1j)+6*np.pi/5))), np.power(np.abs(2+1j), 1/5)*np.imag(np.exp(1j*(np.angle(2+1j)+6*np.pi/5))), 'rx')\n", "plt.plot(np.power(np.abs(2+1j), 1/5)*np.real(np.exp(1j*(np.angle(2+1j)+8*np.pi/5))), np.power(np.abs(2+1j), 1/5)*np.imag(np.exp(1j*(np.angle(2+1j)+8*np.pi/5))), 'rx')\n", "\n", "plt.arrow(0, 0, (np.power(2, 1/5)-0.1)*np.cos(np.angle(2+1j)+0), (np.power(2, 1/5)-0.1)*np.sin(np.angle(2+1j)+0), head_width=0.05, head_length=0.1, fc='r', ec='r')\n", "plt.arrow(0, 0, (np.power(2, 1/5)-0.1)*np.cos(np.angle(2+1j)+2*np.pi/5), (np.power(2, 1/5)-0.1)*np.sin(np.angle(2+1j)+2*np.pi/5), head_width=0.05, head_length=0.1, fc='r', ec='r')\n", "plt.arrow(0, 0, (np.power(2, 1/5)-0.1)*np.cos(np.angle(2+1j)+4*np.pi/5), (np.power(2, 1/5)-0.1)*np.sin(np.angle(2+1j)+4*np.pi/5), head_width=0.05, head_length=0.1, fc='r', ec='r')\n", "plt.arrow(0, 0, (np.power(2, 1/5)-0.1)*np.cos(np.angle(2+1j)+6*np.pi/5), (np.power(2, 1/5)-0.1)*np.sin(np.angle(2+1j)+6*np.pi/5), head_width=0.05, head_length=0.1, fc='r', ec='r')\n", "plt.arrow(0, 0, (np.power(2, 1/5)-0.1)*np.cos(np.angle(2+1j)+8*np.pi/5), (np.power(2, 1/5)-0.1)*np.sin(np.angle(2+1j)+8*np.pi/5), head_width=0.05, head_length=0.1, fc='r', ec='r')\n", "\n", "Drawing_uncolored_circle = plt.Circle( (0, 0), np.power(2, 1/5), fill = False, linestyle = '--')\n", "axes.add_artist( Drawing_uncolored_circle )\n", "\n", "plt.xlim([-2.5,2.5])\n", "plt.ylim([-2.5,2.5])\n", "plt.xlabel('Re(z)')\n", "plt.ylabel('Im(z)')\n", "plt.title( 'Racines cinquièmes de 2+i' )\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exercice 8 : \n", "\n", "Rappeler l'expression de $\\sin(z)$ en fonction de $e^{iz}$ et $e^{-iz}$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$\\sin(z) = \\frac{e^{iz} - e^{-iz}}{2i}$" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "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": "3.9.12" } }, "nbformat": 4, "nbformat_minor": 4 }