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Analysis
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
Given a point z0 where f(z) is either analytic or has an isolated singularity, the residue of f(z) is the coefficient of (z - z0)-1 in the Laurent series expansion of f(z) at z0, or Res(z0)=b1=12πi∫Cf(z)dz. $$ {\text{~Res~}}(z_{0} ) = b_{1} = \frac{1}{{2\pi i}}\mathop \smallint \limits_{C}^{{}} f(z)dz. $$
Complex Representations of Functions
Published in Russell L. Herman, A Course in Mathematical Methods for Physicists, 2013
There are three types of singularities: removable, poles, and essential singularities. They are defined as follows: If f (z) is bounded near z0, then z0 is a removable singularity.If there are a finite number of terms in the principal part of the Laurent series of f (z) about z = z0, then z0 is called a pole.If there are an infinite number of terms in the principal part of the Laurent series of f (z) about z = z0, then z0 is called an essential singularity.
Series
Published in Vladimir Eiderman, An Introduction to Complex Analysis and the Laplace Transform, 2021
To determine the Laurent series expansion of a particular function, we usually use the same methods as used previously for finding a Taylor series expansion: substitution and termwise integration or differentiation, starting with an existing series.
The use of phase portraits to visualize and investigate isolated singular points of complex functions
Published in International Journal of Mathematical Education in Science and Technology, 2019
The Laurent series of a complex function is a representation of that function as a power series which includes terms of negative degree. This topic is usually introduced and studied in a standard course of Complex Analysis. In particular, it may be used to express complex functions in cases where a Taylor series expansion cannot be applied.