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Question
Hi Sir,

  I'm an undergraduate in Math. May I seek your advice to this problem( I'm doing this as a practice for my Complex analysis course):

Let g(z) be an analytic function in punctured ball B(z_1,R)-{z_1} and let N>0 be a fixed non-negative integer such that
lim(z->z_1)g(z)(z-z_1)^m = 0, for any m > N and
lim(z->z_1)g(z)(z-z_1)^n = ∞, for any n > N. Determine the type of singularity of g(z) at z = z_1.

Answer
You must mean n < N or n ≤ N.

What this implies is that there is a Taylor expansion of the function f(z)=g(z)(z-z1)^N around z1. There is also a Laurent series for g(z) that is similar. (Both series converge in some neighborhood.)

There are no singularities here for f(z), and none for g(z) except z1 itself. Clearly z1 is an isolated singularity of g(z), and its order is N. This may also be called a pole.

It's very hard to answer this question in more detail because the "how" and "why" really is just to refer to previous definitions from your text or notes. Ask a follow up question if some of this is not clear.

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