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