Laurent series - Knowlpedia

Statement

If $f$ is holomorphic on an annulus

A=\{z:r<|z-a|<R\},

then there is a unique Laurent series

f(z)=\sum_{n=-\infty}^{\infty}c_n(z-a)^n

converging absolutely and locally uniformly on $A$ . For any positively oriented circle $|\zeta-a|=\rho\subset A$ ,

c_n=\frac{1}{2\pi i}\int_{|\zeta-a|=\rho} \frac{f(\zeta)}{(\zeta-a)^{n+1}}\,d\zeta.

Two-sided convergence

The nonnegative powers form an ordinary power series converging for $|z-a|<R$ ; the negative powers become a power series in $1/(z-a)$ converging for $|z-a|>r$ . Their common annulus is the natural domain of the Laurent expansion.

For $r<\rho<R$ , apply Cauchy's formula on $|\zeta-a|=\rho$ and expand the kernel geometrically on either side of the circle. The resulting integrals are the displayed $c_n$ and converge locally uniformly on the annulus. If two such series agree, integrating their difference against $(z-a)^{-n-1}$ on an intermediate circle gives every coefficient as zero, proving uniqueness.

Principal part

The sum of negative-power terms is the principal part at $a$ . Its pattern classifies an isolated singularity, while $c_{-1}$ is the residue.

References

Lars V. Ahlfors, Complex Analysis, 3rd ed., McGraw–Hill, 1979. Relevant: Chapter 5, §1.