- #1

Nikitin

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Theorem: If a function f(z) has a zero of nth order at z

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 z

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

_{0}, then the function h(z)/f(z) has a pole of order n at z_{0}(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 z

_{0}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..

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