Meromorphic functions and residues

Last time, we discussed Laurent series, which are essentially two-way power series. They are almost as nice as holomorphic functions, but not quite. Maybe we can recoup some of the lost beauty of holomorphicity by imposing a reasonable condition.

We want to allow ourselves to have points where the function isn’t defined, but let’s limit these points. Let’s require them to be isolated points. We say that a function {f} is meromorphic on an open set {U\subseteq {\mathbb C}} if {f} is holomorphic on {U}, except at some number of isolated points. We write {f\in{\mathcal M}(U)}. These isolated points are called poles. We obviously shouldn’t expect {f} to have a power series centered at one of these poles, but it does have a Laurent series.

Let {f\in{\mathcal M}(U)} and let {z_0\in U} be a pole of {f}. Then as we saw last time, {f} has a Laurent series centered at {z_0}:

\displaystyle f(z)=\cdots+a_{-2}(z-z_0)^{-2}+a_{-1}(z-z_0)^{-1}+a_0+a_1(z-z_0)+a_2(z-z_0)^2+\cdots

Moreover, the series has inner radius of convergence 0, so this representation is valid for all {z} close enough to {z_0}.

Now if we take a loop {\gamma} in {U}, and integrate, if {f} has no poles on the interior of {\gamma}, then the integral is zero. This is the Cauchy integral theorem. What if it does have a pole? We can use the generalized Cauchy integral formula we saw last time:

Theorem 1 (Residue theorem) Let {f\in{\mathcal M}(U)} and let {z_0} be a pole of {f} in {U}. Expand {f} as a Laurent series as above. Let {\gamma} be a small counter-clockwise circle about {z_0} such that the only pole in its interior is {z_0}. Then

\displaystyle \displaystyle\frac{1}{2\pi i}\int_\gamma f(z)dz=a_{-1}.

Proof:The generalized Cauchy integral formula we saw last time said that

\displaystyle a_n=\displaystyle\frac{1}{2\pi i}\int_\gamma\frac{f(z)}{(z-z_0)^{n+1}}dz.

Let {n=-1}.

\Box

If we take a loop that goes around several poles, the integral must be the sum of the integrals, as we can build a homotopy as shown in the images below (click to blow up).

The value {a_{-1}} (for an expansion about {z_0}) is called the residue of {f} at the pole {z_0}. In other words, the theorem states that the value of an integral about a contour is the sum of the residues of the poles inside.

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2 Responses to Meromorphic functions and residues

  1. Pingback: A joke « Andy Soffer

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