Uniqueness of Holomorphic Functions

[Wretchosoft]

Homework Statement

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 a_n of points such that lim a_n = a and a_n does not equal a for all n, and if

f'(a_n)/f(a_n)=g'(a_n)/g(a_n)

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

Homework Equations

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

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.

 

[HallsofIvy]

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?

 

[Wretchosoft]

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?

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.