Residue at infinity

In complex analysis, a branch of mathematics, the residue at infinity is a residue of a holomorphic function on an annulus having an infinite external radius. The infinity is a point added to the local space in order to render it compact (in this case it is a one-point compactification). This space noted is isomorphic to the Riemann sphere.[1] One can use the residue at infinity to calculate some integrals.

Definition

Given a holomorphic function f on an annulus (centered at 0, with inner radius and infinite outer radius), the residue at infinity of the function f can be defined in terms of the usual residue as follows:

Thus, one can transfer the study of at infinity to the study of at the origin.

Note that , we have

See also

References

  1. Michèle AUDIN, Analyse Complexe, cursus notes of the university of Strasbourg available on the web, pp. 70–72
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