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Complex valued functions are built into the deﬁnition for Fourier transforms For f: R R, f^(k) = 1 p 2ˇ Z 1 1 e ikxdx; k2R: Fourier transform extends the concept of Fourier series for period functions, is an im-portant tool in analysis and in image and sound processing, and is widely used in elec-trical engineering 3

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An hour on complex numbers Harvard University, 9/23/04, O

• Fourier series: simpliﬁcations • Geometry: i e ﬁnd the length of a di-agonal in a pentagon • Quantum mechanics: wave functions are complex valued, path integrals using imaginary time • Integration like R sin2(x)dx = R (eix − e−ix)2/(2i)2dx • Simplifying trigonometry • Linear algebra: linearization

Complex Analysis - ku

series f(z) = X∞ n=0 f(n)(z 0) n (z−z0)n, for all z in the largest open disc K(z0,ρ) around z0 and contained in G Complex diﬀerentiability is a much stronger requirement than real diﬀeren-tiability because the diﬀerence quotient is required to have one and the same limit independent of the direction from which zapproaches z0 On an

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ANALYSE COMPLEXE Mich`ele Audin R´esum´e Ces notes de cours sont une introduction a l’analyse complexe, avec cent quatre-vingt-onze exercices et vingt-cinq ﬁgures On y ´etablit, pour les fonctions d’une variable complexe, l’´equivalence entre holomorphie et analyticit´e (Cauchy, Morera) On y discute de la question du logarithme

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1 Analyse de Fourier des Signaux d’énergie ﬁnie: Représentation des fonctions par sé-rie,espaces L1 et L2 des signaux, Série de Fourier, Transformation de Fourier, Trans-formation de Laplace Transformée en Z 2 Introduction à la théorie des distributions: Dérivation, convergence, impulsions ap-