Exponential Functions and Their History in Mathematics
Explore the concept of exponential functions, their historical development, equations, and applications in mathematics. Learn about the fundamental properties, derivatives, complex exponential numbers, and series expansions. Dive deeper into the fascinating world of exponentials with this insightful content from Fleurance's JMLL conference in 2016.
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Fleurance, 11/08/16 JMLL/Exponentielle 0
Exponentielle Jean-Marc L vy-Leblond Fleurance, 11/08/16 JMLL/Exponentielle 1
Exponenciel Jean-Marc L vy-Leblond Fleurance, 11/08/16 JMLL/Exponentielle 2
Attention, fil rouge ! Pr requis : Tout savoir d j sur l exponentielle Ou, au moins, conna tre : la notion de fonction la d rivation les nombres complexes (programme de Terminale S) Fleurance, 11/08/16 JMLL/Exponentielle 3
1. Motivations 1. Un brin d histoire 2. L quation fonctionnelle de la fonction exponentielle 1. L quation diff rentielle et sa r solution en s rie enti re 2. Le logarithme naturel 3. La constante e[6 . Bonus : fractions continues] 4. L exponentielle complexe 5. G n ralisation : exponentielles de matrices et d op rateurs 6. S ries de Fourier Fleurance, 11/08/16 JMLL/Exponentielle 4
Le clou de la sance : ii=e-p/2 eip=-1 Fleurance, 11/08/16 JMLL/Exponentielle 5
1.Motivations Fleurance, 11/08/16 JMLL/Exponentielle 6
2. Un brin dhistoire Puissances : carr saxa, cubes axaxa, etc. Chuquet 1484, Stifel 1584 : puissances fractionnaires, n gatives, mais pas de notation adapt e Descartes 1637 : a2, a3 puissances num riques enti res seulement Wallis 1657, Newton 1676, Leibniz 1678 puissances fractionnaires : a1/2= puissances litt rales variables : ap apxaq= ap+q Fleurance, 11/08/16 JMLL/Exponentielle 7
3. Lquation fonctionnelle de la fonction exponentielle multiplication < > addition ? ? f(x) f(y)= f(x+y) Fleurance, 11/08/16 JMLL/Exponentielle 8
4. Lquation diffrentielle et sa r solution en s rie enti re ? f '(x)= lf(x) exp(x)=1+x+x2 +x3 + +xn n!+ 2 6 Fleurance, 11/08/16 JMLL/Exponentielle 9
Le graphe de la fonction exponentielle Fleurance, 11/08/16 JMLL/Exponentielle 10
Fleurance, 11/08/16 JMLL/Exponentielle 11
Les premiers termes de la s rie exponentielle Fleurance, 11/08/16 JMLL/Exponentielle 12
5. La constante e Leonhard Euler, 1727 Leonhard Euler, 1727 Meditatio Meditatio in (M ditation sur des exp riences r centes de tir au canon) in Experimenta Experimenta explosione explosione tormentorum tormentorum nuper nuper instituta instituta Fleurance, 11/08/16 JMLL/Exponentielle 13
Fleurance, 11/08/16 JMLL/Exponentielle 14
e=1 0!+1 1!+1 2!+1 3!+ e = 2.71828182845904523536028747135266249775724709369995 e = = 3 e3= 20,085 Fleurance, 11/08/16 JMLL/Exponentielle 15
Dveloppement de e en fraction continue Fleurance, 11/08/16 JMLL/Exponentielle 16
6. Le logarithme naturel Fleurance, 11/08/16 JMLL/Exponentielle 17
7. Lexponentielle complexe z = x+iy z = z eiargz z y x Fleurance, 11/08/16 JMLL/Exponentielle 18
eip=-1 . Fleurance, 11/08/16 JMLL/Exponentielle 19
Re[exp(x+iy)] =excosy Fleurance, 11/08/16 JMLL/Exponentielle 20
8. Exponentielles de matrices, op rateurs, etc. Fleurance, 11/08/16 JMLL/Exponentielle 21
8. Exponentielles de matrices, op rateurs, etc. Attention la non-commutativit ! Formule de Campbell-Baker-Hausdorff eXeY= X +Y +1 +1 + 2(XY -YX) 12(X2Y -2XYX +YX2) Fleurance, 11/08/16 JMLL/Exponentielle 22
Et, comme cerise sur le gteau, une autre jolie formule reliant e et + dt e-t2 = p - Fleurance, 11/08/16 JMLL/Exponentielle 23
