Analyse Complexe Contents

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Analyse complexe

L’analyse est l’ etude approfondie du calcul di erentiel et int egral Ce cours porte sur le calcul di erentiel et int egral des fonctions complexes d’une va-riable complexe Il s’agit d’un premier cours sur le sujet ou les propri et es des nombres complexes et l’extension aux fonctions de ces nombres des fonctions

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Complex Analysis Lecture Notes

Complex Analysis Lecture Notes Dan Romik About this document These notes were created for use as primary reading material for the graduate course Math 205A: Complex Analysis at UC Davis The current 2020 revision (dated June 15 2021) updates my earlier version of the notes from 2018

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Analyse Complexe

Chapitre 1 Le plan complexe 1 1 Rappels et notations 1 2 La sphère de Riemann* 3 3 Homographies* 4 Exercices 8 Chapitre 2 Fonctions holomorphes d’une variable complexe 11 1 Applications ℝ-linéaires et applications ℂ-linéaires 11 2 Fonctions holomorphes déﬁnition et premières propriétés 13 2 1 Fonctions holomorphes 13 2 2

PDF	Advanced Complex Analysis In this section we develop the maximum princple and related to ideas that lead to compactness of spaces of analytic functions

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Complex Analysis

Contents Preface iii Preface Head iii Acknowledgments iii 1 Complex Numbers 1 De•nitions 1 Algebraic Properties 1 Polar Coordinates and Euler Formula 2 Roots of Complex Numbers 3 Regions in Complex Plane 3 2 Functions of Complex Variables 5 Functions of a Complex Variable 5 Elementary Functions 5 Mappings 7 Mappings by Elementary Functions 8

Qu'est-ce que l'analyse d'une va-Riable complexe ?

L'analyse est l'\u0013etude approfondie du calcul di\u000B\u0013erentiel et int\u0013egral. Ce coursporte sur le calcul di\u000B\u0013erentiel et int\u0013egral des fonctions complexes d'une va-riable complexe.

Comment calculer Integral des fonctions complexes ?

Le calcul int\u0013egral des fonctions complexes est au coeur de leur th\u0013eorie. Il est toujours possible (mais rarement n\u0013ecessaire) de reparam\u0013etrer l'ensemblede ces courbes au moyen d'un seul intervalle [a; b] et d'une fonction contin^umentd\u0013erivable par morceaux.

Comment calculer la derivation par rapport à une variable complexe ?

La d\u0013erivation par rapport \u0012a une variable complexe est formellement iden-tique \u0012a la d\u0013erivation par rapport \u0012a une variable r\u0013eelle. SoientE\u0012Cun ensemble,z0 2Eun de ses points etg: En fz0g !Cune fonction. Les \u0013enonc\u0013es suivants sont alors \u0013equivalents : Pour toute suitefzngn2Nde points deEdistincts dez0, A\u0012 chaque\u000F >0 correspond\u000E >0 tels que

Quelle est la suite d'une fonction complexe ?

Pour toute suitefzngn2Nde points deE, A\u0012 chaque\u000F >0 correspond\u000E >0 tels que Lorsqu'ils sont satisfaits, la fonctionfest ditecontinueenz0. Elle est conti-nue surEsi elle est continue en chaque pointz0 2E. Une fonction complexeest donc continue si et seulement si sa partie r\u0013eelle et sa partie imaginaire lesont toutes les deux.

1.3 Sequences of analytic functions

In this section we develop the maximum princple and related to ideas that lead to compactness of spaces of analytic functions. people.math.harvard.edu

1.6 Harmonic functions

In this section we relate complex analytic functions de ned by the van-ishing of a rst order operator to real harmonic functions, de ned by the vanishing of of a second order operator. Harmonic functions can be de ned on n R and indeed on any Riemannian manifold, and they play a central role in di erential geometry and mathematical physics. Har

c (C)

{ so u and v are smooth, real-valued functions vanishing outside a compact set. Then, by integration by parts, we have Z hru; rvi = Z hu; vi = Z hv; ui: To see this using di erential forms, note that: people.math.harvard.edu

1.7 Additional topics

Here we brie y mention some other classical topics in complex analysis. The Phragmen{Lindelof Theorems. These theorems address the fol-lowing question. Suppose f(z) is an analytic function on the horizontal strip U = fx + iy : a < y < bg, and continuous on U. Can we assert that supU jfj = sup@U jfj? The answer is no, in general. However, the answer

i : Vi C;

with Vi C an open region, such that S i(Vi) = X and the transition functions 1 i j are analytic where de ned. It then makes sense to discuss analytic functions on X, or on any open subset of X: we say f is analytic if f i is analytic for all i. We then obtain a sheaf of rings OX with OX(U) consisting of the analytic maps f : U C. From a more mode

b a

(consisting of matrices satisfying AA = I) acts isometrically on C. b In fact this group is the full group of orientation-preserving isometries. people.math.harvard.edu

Proper maps. A Blaschke product B :

is a rational map of the form d f(z) = ei Y z ai people.math.harvard.edu

3 Entire and meromorphic functions

This section discusses general constructions of functions on C with given ze-ros and poles, and analyzes special cases such as the trigonometric functions and ( z). The study of zeros leads to expressions for entire functions as in nite products, and the study of poles leads to expressions for meromorphic func-tions as in nite sums. people.math.harvard.edu

Z 1 dt

( z) = e ttz : 0 t In other words, ( z) is the Mellin transform of the function e t on R . The Mellin transform is an integral against characters : R C (given by (t) = tz), and as such it can be compared to the Fourier transform (for the group R under addition) and to Gauss sums. Indeed the Gauss sum ( ) = people.math.harvard.edu

4 Conformal mapping

We now turn to the theory of analytic functions as mappings. Here the dominant operation is composition, rather than addition or multiplication. people.math.harvard.edu

Z d Qn 1(

d + ; qi) i where f(qi) = pi. Proof. We will show that: people.math.harvard.edu

Uj 3

Sketch of the proof. Consider a basepoint p in the abstract universal cover : U e U, and let F be the family of all holomorphic maps people.math.harvard.edu

Function

elds. We now return to the case of general elliptic curves. people.math.harvard.edu

(z p1) : : : (z pn)

de nes an elliptic function whenever P ai = P pi. This demonstrates: people.math.harvard.edu

Y(1 qn) 1 X = p(n)qn:

The coe cients power series for Gk( ) involves the function k(n) = people.math.harvard.edu

P dk

djn . The Riemann function (s) arises as the Mellin transform of a theta function, and then modularity translates into the functional equation. people.math.harvard.edu

Analyse complexe (3/26): Fonctions holomorphes (définition)

Analyse complexe (12/26): Intégrales complexe

Analyse complexe (10/26): Intégrales complexes-1

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