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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:

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

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.

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