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Lecture# Complex Analysis: Cauchy Theorem

Description

This lecture covers the Cauchy theorem, complex functions, holomorphic functions, contour integrals, and the interpretation of Cauchy's integral formula. It also explores simply connected domains, residues, and applications of the Cauchy integral formula.

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MATH-201: Analysis III

Calcul différentiel et intégral: Eléments d'analyse vectorielle, intégration par partie, intégrale curviligne, intégrale de surface, théorèmes de Stokes, Green, Gauss, fonctions harmoniques;
Eléments

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Holomorphic function

In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space Cn. The existence of a complex derivative in a neighbourhood is a very strong condition: it implies that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (analytic). Holomorphic functions are the central objects of study in complex analysis.

Analyticity of holomorphic functions

In complex analysis, a complex-valued function of a complex variable : is said to be holomorphic at a point if it is differentiable at every point within some open disk centered at , and is said to be analytic at if in some open disk centered at it can be expanded as a convergent power series (this implies that the radius of convergence is positive). One of the most important theorems of complex analysis is that holomorphic functions are analytic and vice versa.

Cauchy's integral theorem

In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat), is an important statement about line integrals for holomorphic functions in the complex plane. Essentially, it says that if is holomorphic in a simply connected domain Ω, then for any simply closed contour in Ω, that contour integral is zero.

Function of several complex variables

The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space , that is, n-tuples of complex numbers. The name of the field dealing with the properties of these functions is called several complex variables (and analytic space), which the Mathematics Subject Classification has as a top-level heading. As in complex analysis of functions of one variable, which is the case n = 1, the functions studied are holomorphic or complex analytic so that, locally, they are power series in the variables zi.

Residue theorem

In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula. The residue theorem should not be confused with special cases of the generalized Stokes' theorem; however, the latter can be used as an ingredient of its proof.