Complex Analysis: Is my proof of an easy theorem correct?

[Nikitin]

Theorem: If a function f(z) has a zero of nth order at z₀, then the function h(z)/f(z) has a pole of order n at z₀ (where h(z) is analytic at $z_0$).

Can somebody explain this theorem for me? It isn't proved in my book because it's so "easy", but I don't get it? Is the sketch of the proof something like this?

f(z) has a taylor expansion around z₀ which begins on the nth term (since $f^{(1)}(z_0),f^{(2)}(z_0),...,f^{(n-1)}(z_0) = 0$, and thus when you find the taylor expansion of f(z), you divide 1 by said taylor series to gain the expansion of 1/f(z). After splitting up the expansion of 1/f(z) into several different fractions (with the help of some fancy algebra), you will gain a laurent series of order n.

But how do I get to the last part? I am completely confused right now..

 

[Dick]

Theorem: If a function f(z) has a zero of nth order at z₀, then the function h(z)/f(z) has a pole of order n at z₀ (where h(z) is analytic at $z_0$).

Can somebody explain this theorem for me? It isn't proved in my book because it's so "easy", but I don't get it? Is the sketch of the proof something like this?

f(z) has a taylor expansion around z₀ which begins on the nth term (since $f^{(1)}(z_0),f^{(2)}(z_0),...,f^{(n-1)}(z_0) = 0$, and thus when you find the taylor expansion of f(z), you divide 1 by said taylor series to gain the expansion of 1/f(z). After splitting up the expansion of 1/f(z) into several different fractions (with the help of some fancy algebra), you will gain a laurent series of order n.

But how do I get to the last part? I am completely confused right now..

You should also specify that h(z) is nonzero at z0. If f(z) has a zero of nth order at z0, then you can write it as f(z)=(z-z0)^n*g(z) where g(z) is analytic and nonzero at z0. Does that make it clearer?

 

[Nikitin]

Damn, of course! Thanks now I understand the theorem.

Since g(z) is analytic and nonzero, it can be expanded as a taylor series, and thus the largest order negative polynomial possible in the total series is n!