20 - Residue Theorem

Residue Theorem

Suppose that $U$ is an open set in $\mathbb{C}$
Suppose $\gamma$ is a closed curve that is contained in $U$ together with its inside

Suppose $f$ is holomorphic on $U\setminus S$ where $S$ is a finite set of isolated singularities of $f$

Assume that $f$ has no singularity on ${\gamma }^{\ast }$ so that $S\cap {\gamma }^{\ast }=\varnothing$ then

$$\frac{1}{2\pi i}{\int }_{\gamma }f(z)dz=\sum _{a\in S}I(\gamma ,a){\mathrm{Res}}_a(f)$$

Proof

For each $a\in S$ let

$$P_a(f)(z)=\sum _{n=-1}^{-\infty }{c}_n(a)(z-a{)}^n$$

where $P_a(f)(z)$ is the principal part of $f$ at $a$ so is a holomorphic function on $\mathbb{C}\setminus \{a\}$

By definition of $P_a(f)$ then difference $f-P_a(f)$ is holomorphic at $a\in S$ thus

$$g(z)=f(z)-\sum _{a\in S}{P}_a(f)$$

is holomorphic on all of $U$

By Homotopy Cauchy’s Theorem as $U$ contains $\gamma$ and it’s inside then $\gamma$ is null homotopic

$${\int }_{\gamma }g(z)dz=0$$

Hence

$${\int }_{\gamma }f(z)dz=\sum _{a\in S}{\int }_{\gamma }{P}_a(f)(z)dz$$

By Convergence of the principal part then series $P_a(f)$ converges uniformly on ${\gamma }^{\ast }$ so

$$\begin{array}{rl}{\int }_{\gamma }{P}_a(f)dz& ={\int }_{\gamma }\sum _{n=-1}^{-\infty }{c}_n(a)(z-a{)}^n\\ & =\sum _{n=1}^{\infty }\frac{c_{-n}(a)}{(z-a{)}^n}dz\\ & ={\int }_{\gamma }\frac{c_{-1}(a)}{z-a}dz\\ & =2\pi iI(\gamma ,a){\mathrm{Res}}_a(f)\end{array}$$

as for $n>1$ then function $(z-a{)}^{-n}$ has a primitive on $\mathbb{C}\setminus \{a\}$