August 14, 2012 2 Comments
Let be a zero of a meromorphic with multiplicity . Then we can write where . Taking derivatives yields
Hence, for , we get
The residue of this sum is simply the sum of the residues of the two parts. The first term has residue at . The second part has no pole at , and hence zero residue. Thus, the residue of at is .
What if we pick a pole of of ? Then by a similar construction, if is an order pole, we can write and compute
yielding a total residue of .
If a point is neither a pole nor a zero, then is holomorphic at , and has residue zero at .
If we take a big contour around all of these points, then the integral will be the sum of the residues inside the contour, which we have just shown is the number of roots minus the number of poles (counted with multiplicity and order, respectively).
That is, if denotes the number of zeros counted with multiplicity inside a contour , and denotes the number of poles counted with order, then
This result is known as the argument principle.