Laurent series Definition and 157 Threads

Homework Statement Find the Laurent series representation for f(z)=(z-1)/(z-2) at z=i. Homework Equations NA The Attempt at a Solution I have taken multiple derivatives but I keep getting stuck at what to do after I find my representation of my nth derrivative. PLEASE HELP

Homework Statement Find the Laurent series about 0 of 1/sinh up to (and including 0) the z5 term Homework Equations The Attempt at a Solution Since 1/sinh is equal to (1/z) * (1/(1+(z^2/3!)+(z^4/5!)+(z^6/7!)+...)) So if we work on the second term by dividing 1 by denominator and multiply...

Laurent series "throwing away" terms Homework Statement Veeeery similar to https://www.physicsforums.com/showthread.php?p=1868354#post1868354": Determine the Laurent series and residue for f(z) = \frac{1}{(e^{z} - 1)^{z}} Homework Equations We know that the Taylor series expansion...

How does one form a laurent series about the point z0 = 2i for the function: 4cos(z*pi) / (z-z0). Could one take advantage of the power series 1 / (z - z0) 1 / (z - 2i) SUMMATION q^n = SUMMATION (2i)^n = 1 + 2i -4 -8i . . . . . and somehow integrate the rest of the...

We were discussing them in my math methods class today however I'm not really sure how the idea works. Does anyone know of any online references that might be of some help? Google wasn't much help for me =/

why NO multiple Laurent series ?? why are ther Taylor series in several variables (x_{1} , x_{2} ,..., x_{n} but there are NO Laurent series in several variables ? why nobody has defined this series , or why they do not appear anywhere ? i think there are PADE APPROXIMANTS in serveral...

Homework Statement Question is= Find all Laurent series expansion of f(z)=z^4/(3+z^2) around 1. I will be very very thankful if someone can help me to do this question. Homework Equations The Attempt at a Solution can I assume (z-1=u) here and change the function in terms of...

f(x)=\frac{-2}{z-1}+\frac{3}{z+2} our distance is from -2 till 1 we develop around 1 so our distances are 3 and zeo so our areas are 0<|z-1|<3 0<|z-1| 3>|z-1| but i was told to develop around 0<|z-1|<1 there is no such area ?

find the laurent series of f(x)=\frac{-2}{z-1}+\frac{3}{z+2} for 1<|z|<2 i was by my teacher that the radius of convergence is what smaller then the number which makes the denominator 0. if f(x)=\frac{1}{1-z} then the radius is 1 and because 1-1=0 so it is analitical on |z|<1 so...

Homework Statement Calculate a Laurent series about z = 0 for 1/(z2(z+i)) in the region D = {z: |z| < 1} Homework Equations The Attempt at a Solution I used partial fractions to get 1/(z2(z+i)) = 1/z -1/z2 - 1/(z+i) but where do I go from here.

Homework Statement Find the Laurent series for f(z)=1/(z(z-1)) valid on 1<|z-1|<infinity Homework Equations 1/(1+a)=1-a+a^2-a^3... where |a|<1 we are not supposed to use integrals for this problem The Attempt at a Solution I want 1/(z-1) to be in my final answer, so I have...

Homework Statement find the Laurent series for \frac{z+2}{z^{5}-8z^{2}} in 2<|z|<\infty Homework Equations The Attempt at a Solution Well, I factored out z^{5} in the denominator, which left me with a geometric sum (since |z|>2). I've come up with...

Homework Statement f(z) = (z + 2)/(z - 2) a) Find the Maclaurin Series for f on the doman |z| < 2. b) Find the Laurent Series for f centered at z0 = 0 on domain 2 < |Z| < inf. Homework Equations The Attempt at a Solution I'm having a hard time figuring out how (z + 2)/(z -...

[b]1. I am trying to calculate the laurent series expansion of the function 1/(1-z)² in the region 1<|z| [b]2. None [b]3. I can get an answer informally by doing the polynomial division like in high school, but I don't know if this is the right answer and in case it is I cannot prove...

Homework Statement I'm asked to find the Laurent series of some rational function and using partial fractions I encounter something like 1/(c-z)^2 with c > 0. Homework Equations The Attempt at a Solution I've tried several 'algebraic tricks' like multiplying for z^2 or just...

can we define a multivariable power series (laurent series) \sum_{i,j,k,l,...=-\infty}^{\infty}a_{i,j,k,l,...}(X-a)^{i}(Y-b)^{j}(Z-c)^{k}(W-k)^{l}... indices i,j k and l run over ALL the integers positive and negatives how could i calculate the coefficients ?? a_{i,j,k,l} ?

Homework Statement Hi all, I've just calculated the first three nonzero terms of the Laurent series of 1/(cos(z)-1) in the region |z|<2pi, and now I've been asked to 'find the three non-zero central terms of the Laurent expansion valid for 2pi<|z|<4pi' - firstly, what does it mean by...

I don't quite understand a few details here. First, What is the difference between geometric series and laurent series? Than, how do I multiply/divide 2 series with each other? Finally, I have this problem, and I'm really clueless as of what to do. Turn 1/(1-cos(z)) into a laurent series.

The problem: find the laurent series for 1/(z^2-1)^2 valid in 0 < |z-1| < 2 and |z + 1| > 2 we know that f(z) has poles of order 2 at 1 and -1... In the first region, there are no poles (since z=-1 isn't a part of it). We can write the equation as a product of 1/(z-1)^2 and 1/(z+1)^2...

Homework Statement Find the Laurent series expansion of f(z) = (e^z - 1) / (sinz)^3 at z = 0.The Attempt at a Solution Ok, so I'm confused on a number of fronts here. For e^z - 1, I assume you just use the standard power series expansion of e^z and then tack on a -1 at the end, which would...

Homework Statement Hello, I'm having massive troubles with Laurent series'. I'm pretty shocking with series', so I'm probably making some fundamental mistake that you'll want to slap me for, but everytime I try one of these questions I'm wrong ever so slightly. I've attached a couple of...

Hello - I have a problem in general finding the region in which the Laurent series converges... Could someone please help me with this question - I know that this is is meant to be easy (as there is no fully worked solution to this) but I don't understand it: f(z) = 1/ [(z^3)*cosz]...

Find the Laurent Series around z=0 for f(z) = \frac{{2{z^2}}}{{2{z^2} - 5z + 2}} + \sin (\frac{3}{{{z^2}}}), so that the series absolutely converges in z = -i The Attempt at a Solution The singularities of f are z = 0, z = 1/2 and z=2. The Laurent series will converge absolutely in z = -i...

Laurent series (COMPLICATED, URGENT) Homework Statement Find the Laurent series for f(z) = \frac{{2{z^2}}}{{2{z^2} - 5z + 2}} + \sin (\frac{3}{{{z^2}}}) around z = 0, in which the series absolutely converges for z = -i. The Attempt at a Solution I have several problem with this. I...

Homework Statement Find the expression of the Laurent series for f(z) = \frac{z}{{(2z - 1)(\frac{2}{z} - 1)}} so that \sum\limits_{ - \infty }^\infty {{a_n}} \ converges. The Attempt at a Solution First, I find that z = 1/2 and z = 2 (and infinite) are poles of the function f. Then, I...

Homework Statement describe the laurent series for the function f(z) = z^3 cos(\frac {1}{z^2}) b) use your answer to part a to compute the contour integral \int z^3 cos(\frac {1}{z^2}) dz where C is the unit counter-clockwise circle around the origin.Homework Equations The Attempt at a...

Homework Statement http://img243.imageshack.us/img243/4339/69855059.jpg Homework Equations i've heard that the solution requires the use of the exponential taylor series: http://img31.imageshack.us/img31/6163/37267605.jpg The Attempt at a Solution i know that the first step is...

Hi, can u pls help me on this? Homework Statement Find Laurent series that converges for \, 0 < |z - z_0| < R } and determine precise region of convergance \, \frac {1}{z^2 + 1} \,\, Homework Equations The Attempt at a Solution I tried to spilt this into...

Would anyone be willing to check and comment on my work for finding the Laurent series of f(z)=\frac{1+2z}{z^2+z^3} ? Page 1 - http://img23.imageshack.us/img23/7172/i0001.jpg" Page 2 - http://img5.imageshack.us/img5/2140/i0002.jpg" Page 3 -...

I uploaded a scanned page from Schaum's Outline of Complex Variables. I have some questions on how they found the Laurent series in Example 27. http://img19.imageshack.us/img19/7172/i0001.jpg If the image doesn't load, go to - http://img19.imageshack.us/img19/7172/i0001.jpg" In...

Homework Statement find the Laurent series for f(z) = 1/(z(z-1)(z-2)) on the annulus between 1 and 2. with the origin as center. Homework Equations The Attempt at a Solution so i found the partial fraction decomposition of this function and it turns out to be f(z) = 1/2z +...

Homework Statement Hey guys. I need to develop this function into Laurent series. I used the Sin Taylor series and got what I got. Now, is there a trick or something to get the z-2 inside series or is this enough? Thanks. Homework Equations The Attempt at a Solution

Homework Statement Find the Principle part of the Laurent Expansion of f(z) about z=0 in the region 0 < mod z < 1, where f(z) = exp(z) / [(z^2)*(z+1)] Homework Equations 1/(1-z) = Summation (n = 0 to n = infinity) { z^n} The Attempt at a Solution First, by using partial...

Hi, My mathematics professor said that it is possible to construct a Laurent series of sqrt(z) about zero by integrating over a keyhole contour and then taking the limit R --> 0 where R radius of the inner circle. But I think he is mistaken. I don't understand how it is possible to have a...

Homework Statement Find the Laurent series of the function f(z) = Sin(1/(z^2-z)) in the region 0<|z|<infinity. The Attempt at a Solution Now sin(z) = [e^(iz) - e^(-iz)]/(2i) Shall we replace z by 1/(z^2-z) to obtain the Laurent series for f(z)? I tried this but it gets messy. Is...

I'm puzzled by one thing concerning Laurent series. If I have a series, for example f(z) =(z*sinz)/(2z-1) and I'm supposed to make a laurent series of f about the point z=1/2. Now, what would the inner and outer radius of convergence be? I would say that since z=1/2 is a pole, the...

Hi. I am having trouble getting started on this problem. I need to find the Laurent series for: f(z) = exp[(a/2)*(z - 1/z)] in |z|>0. I know that the coefficients are: (1/2pi)*integral[cos(kx) - a*sin(x)]dx |(0 to 2pi) But I am having trouble seeing how to get started on showing that...

Homework Statement We have f(z) = \frac{1}{z^2 - 2z - 3} For this function, we want to find the Laurent Series around z=0, that converges when z=2 and we want to find the area of convergence. Homework Equations \frac{1}{z-3} = -\frac{1}{3}\bigg( \frac{1}{1 - \frac{z}{3}}\bigg) =...

Homework Statement I know the sum of the Laurent series (around x=2) is equal to \frac{1}{x+3} But I can't find what the series is from this information alone. Homework Equations In the textbook, you have (for -1 < x < 1): \frac{1}{1-x} = \sum_{n=0}^{\infty}x^n and for |x|>1 I know...

Laurent series for f(z) = 1/(exp(z)-1)^2 ?? Homework Statement Determine the Laurent series and residue for f(z) = \frac{1}{(e^{z} - 1)^{2}}. Homework Equations We know that the Taylor series expansion of e^{z} is = 1 + z + (z^2)/2! + ... The Attempt at a Solution I am soooo...

I am very confused about how to actually compute a Laurent series. Given an analytic function, we can write down its poles. Then, if I understand correctly, we have to write a Laurent series for each pole. What I'm confused about is the actual mechanics of writing one down. I know that for f(z)...

When doing complex contour integration one can use the C-R formula or the Laurent series and find the first coefficient of the principle part. What are the selection criteria for choosing these methods? Regards, Hob

Hello all, I've got an exam tomorrow so any quick responses would be appreciated. I'm following the Boas section on Laurent series... Anyway, here's my problem: In an example Boas starts with f(z) = 12/(z(2-z)(1+z), and then using partial fractions arrives at f(z) = (4/z)(1/(1+z) +...

[SOLVED] Laurent Series of z/(sin z)^2 Homework Statement Find the first four terms of the Laurent series of f(z) = z/(\sin z)^2 about 0. The attempt at a solution I know that when z = 0, f(z) is undefined so it has a singularity there. This singularity is a pole because \lim_{z \to 0}...

I'm so lost. I can follow the steps in the examples in the book, but all of the examples seem easier than this problem-- and in each case there is a point where you "spot" a similarity to the geometric series or something and then you can just patch it in... But, I just don't know how to make...

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...