Residue (complex analysis) Guide, Meaning , Facts, Information and Description

In complex analysis, the residue is a complex number which describes the behavior of path integrals of a meromorphic function around a singularity. Residues can be computed quite easily and, once known, allow the determination of more complicated path integrals via the residue theorem.

Motivation

As an example, consider the contour integral

where C is some Jordan curve about 0.

Let us evaluate this integral without using standard integral theorems that may be available to us. Now, the Taylor series for e^z is well-known, and we substitute this series into the integrand. The integral then becomes:

Let us bring the 1/z⁵ term into the series, and so, we obtain

The integral now collapses to a much simpler form. Recall

So now the integral around C of every other term not in the form cz⁻¹ becomes zero, and the integral is reduced to

The value 1/4! is known as the residue of e^z/z⁵ at z=0, and is notated as

Calculating residues

Suppose a punctured disk D = {z : 0 < |z − c| < R} in the complex plane is given and f is a holomorphic function defined (at least) on D. The residue Res(f, c) of f at c is the coefficient a₋₁ of (z − c)⁻¹ in the Laurent series expansion of f around c. This coefficient can often be computed by combining several known Taylor series. At a simple pole, the residue is given by:

According to the integral formula given in the Laurent series article we have:

where γ traces out a circle around c in a counterclockwise manner. We may choose the path γ to be a circle in radius ε around c were ε is as small as we desire.

To calculate the residue of a function around z = c, a pole of order n, one may use the following formula:

If the function f can be continued to a holomorphic function on the whole disk { z : |z − c| < R }, then Res(f, c) = 0. The converse is not generally true.

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