4.4 Further Properties of Meromorphic and Entire Functions
Theorem 4.4.1.
Let be a region and be meromorphic. Let be a positively oriented Jordan curve that is null-homotopic in . If has no zeros on , then has finitely many zeros and poles in the region bounded by . Denote the zeros of in the bounded region by with respective multiplicities , and the poles by with respective orders . Let be any function holomorphic on a neighborhood of the closure of the bounded region. Then
Proof.
Choose disks with pairwise disjoint closures around each zero and around each pole , with sufficiently small so that these disks are contained in , disjoint from , and contained in the neighborhood where is holomorphic. The function
is holomorphic on
since is holomorphic there, is meromorphic with no other singularities, and on . The oriented boundary of this punctured domain is . By Cauchy–Goursat (Theorem 3.1.7),
Thus,
Near each zero , write where is holomorphic at with . Then
so
Then,
where the first term has been reduced by the Cauchy–Goursat Formula (Theorem 3.1.8) and the second integral vanishes by the Cauchy–Goursat Theorem (Theorem 3.1.7).
Near a pole , write where is holomorphic at with . Then
so
A similar calculation yields that
Combining these,
Theorem 4.4.2 (Argument Principle).
Let be a region and be meromorphic. Let be a simple, closed, positively oriented curve that is null-homotopic in . If has no zeros or poles on , then has finitely many zeros and poles in the region bounded by , and the number of zeros, , minus the number of poles, , counting multiplicities and orders, is given by
Let be the image of under the map . Then .