Laurent series Definition and 162 Threads

Homework Statement Write TWO laurent series in powers of z that represent the function f(z)= \frac{1}{z(1+z^2)} In certain domains, and specify the domains Homework Equations Well that's my prob, not sure what the terms in the Laurent series are The formula I'm looking at is...

I understand perfectly well how to do Taylor series, but I am foggy on these Laurent series. Say, we have something like, f\left( z \right)\; =\; \frac{1}{z^{2}\cdot \sin \left( z \right)} I think I need to use the taylor series expressions for sin(z) but otherwise, I am not sure what to...

Say we have the function: \frac{1}{\left( z-1 \right)\left( z+2 \right)} Using partial fractions, \frac{1}{\left( z-1 \right)\left( z+2 \right)}\; =\; \frac{1}{z-2}\; -\; \frac{1}{z\; -\; 1} My question comes in on why and how these equations are manipulted for different regions. Now for a)...

Homework Statement Let D be a subset of C and D is open. Suppose a is in D and f:D\{a} -> C is analytic and injective. Prove the following statements: a) f has in a, a non-essential singularity. b) If f has a pole in a, then it is a pole of order 1. c) If f has a removable singularity...

This is more of a general question than a specific question. ---------- Find the annulus of convergence for the Laurent series \sum_{n=-\infty}^{-1} \left( \frac{z}{2} \right)^n + \sum_{n=0}^\infty \frac{z^n}{n!} -------- I know what to do for the second series, but I am not sure about the...

How does one integrate \int_{}^{} \frac{e^x}{x}dx I could expand it using a Laurent series and than integrating term by term but are there more elementary methods?

Define: \mathbb{C}((t)) = \{t^{-n_0}\sum_{i=0}^{\infty}a_it^i\ :\ n_0 \in \mathbb{N}, a_i \in \mathbb{C}\} What is its algebraic closure? My notes say that it is "close" to: \bigcup _{m \in \mathbb{N}}\mathbb{C}((t))(t^{1/m}) where \mathbb{C}((t))(t^{1/m}) is the extention of the...

Just wondering where to go with this one.. calculate the laurent series of \frac{1}{e^z-1} don't even know where to start on it I know e^z={{\sum^{\infty}}_{j=0}}\frac{z^j}{j!} but not much else...

Hi all,, I have an Exam tommorow and this question is irritating me...Pls help Laurent series of the function f(z)=1/z^2 for |z-a|>|a| .a is not equal to zero... I am waiting for yours responses...I will be highly thankful to you.

I need help with a problem from Complex Analysis. The directions say find the Laurent series that converges for 0<|z|<R and determine the precise region of convergence. The expression is : e^z/(z-z^2). I understand how to do the other 7 problems in this section but not this one. Can someone...

My question is about the coefficients of a complex laurent series. As far as I know, there are three kinds of series: those which converge in a finite circular region around the expansion point z0,(aka taylor series), those that converge in a ring shaped region between two circles centered at...

Does anyone know of any examples of the explicit calculation of the Laurent series of a complex function? Any information would be appreciated.