# Composition of meromorphic functions

1. Mar 3, 2013

### fortissimo

1. The problem statement, all variables and given/known data

If f is meromorphic at a in ℂ^ (extended complex plane), and g is meromorphic at b = f(a) in ℂ^, show that h := g ° f is meromophic at a, with an exception. If f takes on the value b in ℂ^ with multiplicity m at a and g assumes the value c = g(b) in ℂ^ with multiplicity n at b, what is the multiplicity of c at a?

2. Relevant equations

3. The attempt at a solution

For the first part I don't know exactly what to do. Could you use the fact that f and g can be expanded into Laurent series?
For the second part, at least if a, b, c ≠ ∞, we have that g(f(z)) - c = g1(f(z))*(f(z) - b)^n where g1(b) ≠ 0, and f(z) - b = f1(z)*(z-a)^m (where f1(a) ≠ 0), so the multiplicity is nm. It seems tedious if you have to make different cases for a, b, c = ∞. Or do you need to do this?