# Théorème intégral de Cauchy

{style d'affichage entier _{C}F(z),dz=0.} Contenu 1 Déclaration 1.1 Fundamental theorem for complex line integrals 1.1.1 Formulation on Simply Connected Regions 1.1.2 General Formulation 1.1.3 Main Example 2 Discussion 3 Preuve 4 Voir également 5 Références 6 External links Statement Fundamental theorem for complex line integrals If f(z) is a holomorphic function on an open region U, et {gamma de style d'affichage } is a curve in U from {style d'affichage z_{0}} à {style d'affichage z_{1}} alors, {style d'affichage entier _{gamma }F'(z),dz=f(z_{1})-F(z_{0}).} Aussi, when f(z) has a single-valued antiderivative in an open region U, then the path integral {style de texte entier _{gamma }F'(z),dz} is path independent for all paths in U.

Formulation on Simply Connected Regions Let {displaystyle Usubseteq mathbb {C} } be a simply connected open set, et laissez {style d'affichage f:Uto mathbb {C} } être une fonction holomorphe. Laisser {gamma de style d'affichage :[un,b]to U} be a smooth closed curve. Alors: {style d'affichage entier _{gamma }F(z),dz=0.} (The condition that {style d'affichage U} be simply connected means that {style d'affichage U} has no "holes", or in other words, that the fundamental group of {style d'affichage U} is trivial.) General Formulation Let {displaystyle Usubseteq mathbb {C} } be an open set, et laissez {style d'affichage f:Uto mathbb {C} } être une fonction holomorphe. Laisser {gamma de style d'affichage :[un,b]to U} be a smooth closed curve. Si {gamma de style d'affichage } is homotopic to a constant curve, alors: {style d'affichage entier _{gamma }F(z),dz=0.} (Recall that a curve is homotopic to a constant curve if there exists a smooth homotopy from the curve to the constant curve. Intuitivement, this means that one can shrink the curve into a point without exiting the space.) The first version is a special case of this because on a simply connected set, every closed curve is homotopic to a constant curve.

Main Example In both cases, it is important to remember that the curve {gamma de style d'affichage } does not surround any "holes" in the domain, or else the theorem does not apply. A famous example is the following curve: {gamma de style d'affichage (t)=e^{it}quad tin left[0,2pi right],} which traces out the unit circle. Here the following integral: {style d'affichage entier _{gamma }{frac {1}{z}},dz=2pi ineq 0,} is nonzero. The Cauchy integral theorem does not apply here since {style d'affichage f(z)=1/z} is not defined at {displaystyle z=0} . Intuitivement, {gamma de style d'affichage } surrounds a "hole" in the domain of {style d'affichage f} , alors {gamma de style d'affichage } cannot be shrunk to a point without exiting the space. Ainsi, the theorem does not apply.

Discussion As Édouard Goursat showed, Cauchy's integral theorem can be proven assuming only that the complex derivative {style d'affichage f'(z)} exists everywhere in {style d'affichage U} . This is significant because one can then prove Cauchy's integral formula for these functions, and from that deduce these functions are infinitely differentiable.

The condition that {style d'affichage U} be simply connected means that {style d'affichage U} has no "holes" ou, in homotopy terms, that the fundamental group of {style d'affichage U} est trivial; par exemple, every open disk {style d'affichage U_{z_{0}}={z:la gauche|z-z_{0}droit|

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