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Question
Hi Sir,

  May I seek your advice to this problem:

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
Assume g(z) = (z-z1)^(-p)・f(z) where f(z) is analytic everywhere, especially at z=z1. Then (I'll leave out writing the limits z->z1)

(z-z1)^(m-p)・f(z1) = 0
(z-z1)^(n-p)・f(z1)  = ∞

from which we can say m-p ≥ 1 and n-p≤-1. Rearranging we get

0 ≤ p ≤ m-n-2

so the singularity would appear to be a pole of order p. I'm a little concerned about the statement "for any m > N and n > N" since we also have the contraint

2 ≤ m-n

but hopefully this is not a showstopper. If it is, please let me know.

Randy

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randy patton

Expertise

college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

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26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

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J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

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M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

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