Lecture

Complex Analysis: Laurent Series and Residue Theorem

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Description

This lecture covers the completion of the chapter on Laurent series, focusing on the residue theorem and its applications in complex integral calculus. The instructor begins by reviewing the previous results related to determining the order of poles in functions expressed as fractions. The lecture explains how to identify poles and calculate residues using Laurent series expansions. The instructor provides a detailed example involving the function f(z) = 1/(z sin z), analyzing its behavior around the pole at zero. The process of isolating the residue is demonstrated through a systematic approach of eliminating singularities and deriving necessary terms. The lecture transitions into the definition of complex integration, emphasizing the importance of parametrization of curves in the complex plane. The instructor introduces the residue theorem, illustrating how to compute integrals of holomorphic functions around singularities. The session concludes with practical examples, reinforcing the concepts of residues and their significance in evaluating complex integrals, setting the stage for further exploration in complex analysis.

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Ontological neighbourhood

Mathematics

Analysis: Complex analysis

Related concepts (32)

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.

Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.

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.

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.

Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if its Taylor series about converges to the function in some neighborhood for every in its domain.

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