Meromorphic functions and residues
August 7, 2012 2 Comments
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 is meromorphic on an open set if is holomorphic on , except at some number of isolated points. We write . These isolated points are called poles. We obviously shouldn’t expect to have a power series centered at one of these poles, but it does have a Laurent series.
Let and let be a pole of . Then as we saw last time, has a Laurent series centered at :
Moreover, the series has inner radius of convergence 0, so this representation is valid for all close enough to .
Now if we take a loop in , and integrate, if has no poles on the interior of , 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 and let be a pole of in . Expand as a Laurent series as above. Let be a small counter-clockwise circle about such that the only pole in its interior is . Then
Proof:The generalized Cauchy integral formula we saw last time said that
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 (for an expansion about ) is called the residue of at the pole . 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.