  ## Encyclopedia > Residue (complex analysis)

Article Content

# Residue (complex analysis)

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.

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 of f at c, written as Res(f, c), is then defined as 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; it is also possible to use the integral formula given in the Laurent series article:

$\operatorname{Res}(f,c) = {1 \over 2\pi i} \int_\gamma f(z)\,dz$

where γ traces out a circle around c in a counter clockwise manner.

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.

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 French resistance ... to threat of paid informants, there was also Milice[?], collaborating Vichy France police force lead by Joseph Darnand[?]. Its methods were as unpleasant as those of ...  