Integral of Cauchy

Introduction

This integral establishes the link between the Intégrale of Riemann, traditional but with real variables, and the variable complexes.

Chemin

A curve in X is a continuous application $\ gamma: \ alpha, \ beta \ rightarrow X, \ alpha < \ beta \ in \ mathbb {R}$ . One calls $\ alpha, \ beta$ the interval of parameter setting of γ, and one notes $\ gamma^ {*}$ the image of the application.

If $\ gamma (\ alpha) = \ gamma (\ beta) \,$ , the curve known as is closed.

A way γ is a curve of the complex plan provided with its Euclidean topology, continuously derivable by pieces.

A closed loop is a closed curve which is also a way.

Definition of the integral

By considering a way γ, and $f: \ gamma^ {*} \ rightarrow \ mathbb {C}$ , a continuous function, one defines the integral of Cauchy of F on the way γ like this:

$\ int_ {\ gamma} F (Z) dz = \ int_ {has} ^ {B} F (\ gamma (T)) \ gamma' (T) dt$

This integral is well defined within the meaning of Riemann.

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