On Physical Lines of Force

2014-01-29T15:32:34Z

117.194.89.46:

In [[complex analysis]] a branch of mathematics, the '''residue at infinity''' is a [[Residue (complex analysis)|residue]] of a [[holomorphic function]] on an [[Annulus (mathematics)|annulus]] having an infinite external radius. The ''infinity'' <math>\infty</math> is a point added to the local space <math>\mathbb C </math> in order to render it [[compact space|compact]] (in this case it is a [[Alexandroff extension|one-point compactification]]). This space noted <math> \hat{\mathbb C} </math> is [[isomorphism|isomorphic]] to the [[Riemann sphere]].<ref>Michèle AUDIN, ''Analyse Complexe'', cursus notes of the university of Strasbourg [http://www-irma.u-strasbg.fr/~maudin/analysecomp.pdf available on the web], pp. 70–72</ref> One can use the residue at infinity to calculate some [[integral]]s.

==Definition==
Given a holomorphic function ''f'' on an [[Annulus (mathematics)|annulus]] <math> A(0, R, \infty) </math> (centered at 0, with inner radius <math>R</math> and infinite outer radius), the '''residue at infinity''' of the function ''f'' can be defined in terms of the usual [[residue (mathematics)|residue]] as follows:

:<math> \mathrm{Res}(f,\infty) = \mathrm{Res}\left( {-1\over z^2}f\left({1\over z}\right), 0 \right)</math>

Thus, one can transfer the study of <math> f(z) </math> at infinity to the study of <math> f(1/z) </math> at the origin.

Note that <math>\forall r > R</math>, we have

:<math> \mathrm{Res}(f, \infty) = {-1\over 2\pi i}\int_{C(0, r)} f(z) \, dz</math>

== See also ==
* [[Riemann sphere]]
* [[Algebraic variety]]
* [[Residue theorem]]

== References ==
{{Translation/Ref|fr|Résidu à l'infini|oldid=59523358}}
<references/>
* Murray R. Spiegel, ''Variables complexes'', Schaum, ISBN 2-7042-0020-3
* [[Henri Cartan]], ''Théorie analytique des fonctions d'une ou plusieurs varaiables complexes'', Hermann, 1961

[[Category:Complex analysis]]

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