# Complex Analysis problem

1. Mar 25, 2009

### Wretchosoft

1. The problem statement, all variables and given/known data Let f and g be two holomorphic functions in a connected open set D of the plane which have no zeros in D; if there is a sequence an of points such that lim an = a and an does not equal a for all n, and if

f'(an)/f(an)=g'(an)/g(an)

show that there is a constant c such that f=cg in D.

2. Relevant equations If f is identically zero at a point in a connected open set, then f is identically zero on the whole set.

3. The attempt at a solution I have shown that (f/g)'(a) = 0, but I don't see how that would imply that the derivative is identically zero at that point.

Last edited: Mar 25, 2009
2. Mar 25, 2009

### HallsofIvy

Staff Emeritus
I don't know what you mean by "identically 0 at a point"! "Identically" zero means zero at every point of some set. Are you sure you have quoted your "relevant equation" correctly?

3. Mar 25, 2009

### Wretchosoft

Ah, sorry. What I mean is that the derivatives of f of every order are 0 at that point, so that f is equal to the zero function at that point. I may be abusing the term.