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Residue at infinity

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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 ∞ {\displaystyle \infty } is a point added to the local space C {\displaystyle \mathbb {C} } in order to render it compact (in this case it is a one-point compactification). This space denoted C ^ {\displaystyle {\hat {\mathbb {C} }}} is isomorphic to the Riemann sphere. One can use the residue at infinity to calculate some integrals.

Definition

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

Res ⁡ ( f , ∞ ) = − Res ⁡ ( 1 z 2 f ( 1 z ) , 0 ) {\displaystyle \operatorname {Res} (f,\infty )=-\operatorname {Res} \left({1 \over z^{2}}f\left({1 \over z}\right),0\right)}

Thus, one can transfer the study of f ( z ) {\displaystyle f(z)} at infinity to the study of f ( 1 / z ) {\displaystyle f(1/z)} at the origin.

Note that ∀ r > R {\displaystyle \forall r>R} , we have

Res ⁡ ( f , ∞ ) = − 1 2 π i ∫ C ( 0 , r ) f ( z ) d z {\displaystyle \operatorname {Res} (f,\infty )={-1 \over 2\pi i}\int _{C(0,r)}f(z)\,dz}

Since, for holomorphic functions the sum of the residues at the isolated singularities plus the residue at infinity is zero, it can be expressed as:

Res ⁡ ( f ( z ) , ∞ ) = − ∑ k Res ⁡ ( f ( z ) , a k ) . {\displaystyle \operatorname {Res} (f(z),\infty )=-\sum _{k}\operatorname {Res} \left(f\left(z\right),a_{k}\right).}

Motivation

One might first guess that the definition of the residue of f ( z ) {\displaystyle f(z)} at infinity should just be the residue of f ( 1 / z ) {\displaystyle f(1/z)} at z = 0 {\displaystyle z=0} . However, the reason that we consider instead − 1 z 2 f ( 1 z ) {\displaystyle -{\frac {1}{z^{2}}}f\left({\frac {1}{z}}\right)} is that one does not take residues of functions, but of differential forms, i.e. the residue of f ( z ) d z {\displaystyle f(z)dz} at infinity is the residue of f ( 1 z ) d ( 1 z ) = − 1 z 2 f ( 1 z ) d z {\displaystyle f\left({\frac {1}{z}}\right)d\left({\frac {1}{z}}\right)=-{\frac {1}{z^{2}}}f\left({\frac {1}{z}}\right)dz} at z = 0 {\displaystyle z=0} .

See also

References

  1. Michèle Audin, Analyse Complexe, lecture notes of the University of Strasbourg available on the web, pp. 70–72
  • Murray R. Spiegel, Variables complexes, Schaum, ISBN 2-7042-0020-3
  • Henri Cartan, Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes, Hermann, 1961
  • Mark J. Ablowitz & Athanassios S. Fokas, Complex Variables: Introduction and Applications (Second Edition), 2003, ISBN 978-0-521-53429-1, P211-212.
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