You are here:

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.

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.

Volunteer

#### Clyde Oliver

##### Expertise

I can answer all questions up to, and including, graduate level mathematics. I am more likely to prefer questions beyond the level of calculus. I can answer any questions, from basic elementary number theory like how to prove the first three digits of powers of 2 repeat (they do, with period 100, starting at 8), all the way to advanced mathematics like proving Egorov's theorem or finding phase transitions in random networks.

##### Experience

I am a PhD educated mathematician working in research at a major university.

Organizations
AMS

Publications
Various research journals of mathematics. Various talks & presentations (some short, some long), about either interesting classical material or about research work.

Education/Credentials
BA mathematics & physics, PhD mathematics from a top 20 US school.

Awards and Honors
Various honors related to grades, various fellowships & scholarships, awards for contributions to mathematics and education at my schools, etc.

Past/Present Clients
In the past, and as my career progresses, I have worked and continue to work as an educator and mentor to students of varying age levels, skill levels, and educational levels.