A property of meromorphic functions (?)

[evilcman]

Is this statement true: "If two meromorphic functions have the same poles(all simple) and the same
zeros(all simple), than they are proportional."? If it is true, than why? Thanks for the help...

 

[Wizlem]

I believe all meromorphic functions can be written as the ratio of two holomorphic functions. A holomorphic function can be written as a (possibly infinite) product of monomials which are of the form (x-a) where a is a zero of the function. So if f(x) is meromorphic we can write it as g(x)/h(x) where the zeros of g(x) are the zeros of f(x) and the zeros of h(x) are the poles of f(x). If two functions have the same zeros and poles their g(x) and h(x) can only differ by a multiplicative constant.

 

[evilcman]

All meromorphic functions can be written as the ratio of two holomorphic functions, that is true.

The second statement is not true. In general a holomorphic function can't be written as a product of monomials.
You will in general also have an exponential in it. And the exponential in it can have a holomorphic function in the
argument: http://en.wikipedia.org/wiki/Weierstrass_factorization_theorem

So is you have the meromorphic functions f1(x)/g1(x) and f2(x)/g2(x), where the fs and gs are holomorphic,
than for example you could have for example f1(x) = exp(h(x)) f2(x) with h(x) some holomorphic function.

 

[FactChecker]

What about e^z and e^2*z? Is that a counterexample?