On Physical Lines of Force

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 $\mathrm{\infty }$ is a point added to the local space $\mathbb{ℂ}$ in order to render it compact (in this case it is a one-point compactification). This space noted $\widehat{\mathbb{ℂ}}$ 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 $A(0,R,\mathrm{\infty })$ (centered at 0, with inner radius $R$ and infinite outer radius), the residue at infinity of the function f can be defined in terms of the usual residue as follows:

$\mathrm{R}\mathrm{e}\mathrm{s}(f,\mathrm{\infty })=\mathrm{R}\mathrm{e}\mathrm{s}(\frac{-1}{z^2}f\left(\frac{1}{z}\right),0)$

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

Note that $\mathrm{\forall }r>R$, we have

$\mathrm{R}\mathrm{e}\mathrm{s}(f,\mathrm{\infty })=\frac{-1}{2\pi i}\underset{C(0,r)}{\int }f(z)dz$

See also

• Riemann sphere
• Algebraic variety
• Residue theorem

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