Laurent series Definition and 162 Threads

Homework Statement Specifically, I'm trying to find the laurent series for f(z)=\frac{z^2}{z+1} around the point z=-1. My real problem is my procedure in general though. I'm not sure what I'm doing wrong on a lot of these Laurent Series but for some reason I'm struggling with them. (Even...

Hi! There's a few things I'm confused about, and I hope some of you would bother helping me with: 1) Why do I need these laurent series? As I understood from Calculus 1, the taylor series around ##x_0## will always approximate a function ##f(x)## gradually better as the order ##n## increases...

Laurent series at infinity point I already calculated it, but my work was too long, I really wish to find a shorter route. Calculate the Laurent series of \frac{1}{(z^{2}+1)^{2}} around z_{0} = 0 First, I used simple fractions and I got: \frac{1}{(z^{2}+1)^{2}}=\frac{-i}{4} \frac{1}{z-i} +...

Homework Statement Find the Laurent expansion for \frac{1}{z^2-1} in the annulus 1 < |z-2| < 3 The Attempt at a Solution I've gotten to the last parts but getting stuck there. First I expanded the denominator and did a partial fraction decomposition and arrived at...

Homework Statement Consider a series of three charges arranged in a line along the z-axis, charges +Q at z = D and charge -2Q at z = 0. (a) Find the electrostatic potential at a point P in the x, y-plane at a distance r from the center of the quadrupole. (b) Assume r >> D. Find the...

Please refer to attached image. Hi, I'm a bit lost here with the first question. Unfortunately the online lecture covering this material isn't available due to their having been made some technical difficulties, and I find our textbook difficult to comprehend! My lecture notes are pretty...

In general I am rather confused by this type of problem. The textbook has a single example and does not show (m)any of its steps so I'm lost. I have a test this coming Thursday and the following is the only question of this type that the prof. has recommended: "23. Use equations (12) and (13)...

"[F]ind the Laurent series for the given function about the indicated point. Also, give the residue of the function at the point. 8. ##\frac{z^{2}}{z^{2} -1}##; ##z_{o} = 1##" (Complex Variables, 2nd edition; Stephen D. Fisher, pg. 150) I'm not very comfortable with Laurent series (or...

Hey everyone, I've got a question that might be interesting to those of you that enjoy maths. Note: I only did a physics degree, so I have never really done analysis in the proper way a mathematician has. But I am eager to learn about more rigorous maths. Right, so for a Laurent series about a...

Homework Statement Find the Laurent series for the function that converges at 0 < \left|z-z_0\right| < R Homework Equations \frac{z^2-4}{z-1} z_0 = 1The Attempt at a Solution I think this is going to be a finite sum. I think it could be wrong, though, because it certainly differs with the...

Find the Laurent series of the following function in a neighborhood of the singularly indicated, and use it to classify the singularity. Homework Statement f(z) = \frac{1}{z^2-4} ; z_0=2 Homework Equations Laurent series \sum_{-\infty}^{\infty} a_n (z-c)^n The Attempt at a Solution I...

Homework Statement Determine the Laurent series expansion of \frac{1}{e^z - 1} The attempt at a solution I've spotted that \frac{1}{e^z - 1} = \frac{1}{2}\left( \coth{\frac{z}{2}} - 1\right) but I don't know what to do next. WolframAlpha gives the series centred at 0 as...

Homework Statement Find the Laurent series for the given function about the specified point. Also, give the residue of the function at the point. $$ \frac{z^2}{z^2 - 1}, z_0 = 1 $$ Homework Equations A Laurent expansion is comparable to a power series, except that it includes negative...

Homework Statement f(z) = \frac{1}{ \exp{ \frac {z^2 - \pi/2}{ \sqrt{3} } } + i } Find the residue of f(z) at z_0 = \frac{ \sqrt(\pi) }{2 } ( \sqrt(3) - i ) Homework Equations The Attempt at a Solution I was able to verify that the given z_0 is a singularity, and...

Homework Statement It f is a meromorphic function with finite number of singularities, prove that the the principal part of the laurent series centered at a singularity has infinite convergence radius. Homework Equations f(z)=Ʃ(a_n)(z-z_j) where z_j is the singularity. Principal part =...

My task is to solve the integral \frac{1}{\cos 2z} on the contour z=|1| using a Laurent series. The easy part of this is the geometric part. I drew a picture of the problem. I see that there are two singularity points that occur within the contour region at \pm \frac{\pi}{4}. I realize...

Say we have a simple function like f(z)=4z/[(z-1)(z-3)2] I'll use this example to demonstrate my undertanding of the motivation behind and usefulness of Laurent series: if we examine f(z), we see it is analytic except where z = 1 and where z = 3, which means it can expanded in a Taylor...

Hey all, I am doing a Schwarz-Christoffel transformation and I am trying to calculate the integral analytically using the residue theorem. My integral is the following: \int^\zeta _{\zeta_0} (z+1)\frac{1}{(z+2.9)^{{b_1}/\pi}{(z-0.5)^{{b_2}/\pi}}}dz This has two poles at -2.9 and 0.5. b_1 and...

Homework Statement Find a Laurent Series of f(z)=\frac{1}{(2z-1)(z-3)} about the point z=1 in the annular domain \frac{1}{2}<|z-1|<2. Homework Equations The Attempt at a Solution By partial fraction decomposition...

Homework Statement Find the laurent series about z=-2 for: f(z) = 1/(z(z+2)3) Homework Equations The Attempt at a Solution Setting t = z+2 yields: f(t) = 1/(t3(t-2)) = 1/t (-1/(2(1-t/2))) = (1/t)3 * (-1/2) * Ʃ(t/2)n which can be put together in a sum, but I can't be...

Homework Statement Calculate the laurent series expansion about he points specified, classify the singularity and sate the region of convergence for. \frac{1}{z^2 - 1} at (i) z=1 (ii) z=-1 (iii)z=0 Homework Equations The Attempt at a Solution \frac{1}{z^2 - 1} =...

Use geometric series to find the Laurent series for f (z) = z / (z - 1)(z - 2) in each annulus (a) Ann(1,0,1) (b) Ann(1,1,∞) Ann(a,r,R) a= center, r=smaller radius, R=larger radius Ann(1,0,1)=D(1,1)\{0} My attempt: f(z)= -1/(z-1) + 2/(z-2) geometric series: Σ[n=0 to inf] z^n - 1/2...

Suppose we have f(z) = \sum_{n=-\infty}^\infty c_n z^n; \quad U(z) = \sum_{n=0}^\infty c_n z^n; \quad L(z) = \sum_{n=1}^\infty c_{-n} z^{-n} where f(z) converges* in the interior of some annulus with inner radius r and outer radius R > r. Further suppose U(z) has radius of convergence R_0...

Find the Laurent series of the form $\sum\limits_{n = -\infty}^{\infty}c_nz^n$ for $f(z) = \dfrac{z^2}{(z - 1)(z - 3)}$ that converges in an annulus containing the point $z = 2$, and state precisely where this Laurent series converges.} By the method of partial fractions, (how does the 1...

Homework Statement Find the Laurent series of \frac{(z+2)}{(z-1)} on C_1: 1 < |z| and C_2: 0 < |z| < 1 Homework Equations I have a formula for computing Laurent series, but it includes an integral that is impossible to solve. For everything that I've read, no one actually...

I am trying to understand the idea of annulus of convergence. This is the example I have been looking at but it has me completely stumped. [∞]\sum[/n=1] (z^n!)(1-sin(1/2n))^(n+1)! + [∞]\sum[/n=1] (2n)!/[((n!)^2)(z^3n)] All of the examples I have worked on in the past have been...

Homework Statement 1. Evaluate \int_{c_{2}(0)} f(z)dz = \int_{c_{2}(0)} \frac{z^{m}}{1+z^{3}}dz Where c_{2}(0) is the circle of radius 2 centered at the origin with positive orientation (ccw). I have done the question myself and compared it with the solution. However, I don't think I am...

Homework Statement I have some past exam questions that I am confused with Homework Equations a_{n} = \frac{1}{2\pi i} \oint_\gamma \frac{f(z)}{z-a}\, dz The Attempt at a Solution I'm not sure how to approach this, I'm completely lost and just attempted to solve a few: a) it says f(z)...

Homework Statement expansion of e^z/z*(1-z) Homework Equations The Attempt at a Solution

Find the Laurent series for $1/z^2(1 - z)$ in the regions $0 < |z| < 1$ $$ \frac{1}{z^2(1 - z)} = \frac{1}{z^2}\frac{1}{1-z} $$ Since $|z| < 1$, the geometric series will converge so $$ \frac{1}{z^2}\sum_{n = 0}^{\infty}z^n = \sum_{n = -2}^{\infty}z^n. $$$|z| > 1$ The geometric series will...

I have never done a Laurent series nor have we went over it in class but I guess I am supposed to know it perfectly already. The explanation in the book isn't that great. Find the Laurent series of the form $\sum\limits_{n = -\infty}^{\infty}c_nz^n$ for $f(z) = \dfrac{33}{(2z - 1)(z + 5)}$...

I wonder why z^(-1/2) cannot be expanded in Laurent series with center z=0. Anyone knows?

The textbook used in one of my courses talks about expanding functions in powers of 1/z aka negative powers of z. The problem is that I cannot recall that any previous course taught me/challenged me on how to expand functions in negative powers. For example, Taylor series only have positive...

Homework Statement Obtain the first few terms of the Laurent series for the following function in the specified domain: \frac{1}{e^z-1} for 0 < |z| < 2\pi. Homework Equations The Attempt at a Solution I've attempted a few approaches, but haven't really gotten anywhere. For...

Hello, I am having difficulty matching one term in my Laurent series to that which mathematica tells me is the correct answer. For the function f(z)=log\frac{1+z}{1-z} we know that there exists a k such that Log|1+z|-Log|1-z|+i2\pi k Now, we know that the Taylor series of f is as...

Homework Statement Find the Laurent series of f(z) = exp(1/z)/(z-1) around 0, and find Res{f(z), 0}.Homework Equations The Attempt at a Solution To find the Laurent series, I wrote exp(1/z) = \sum_{n=0}^∞ z-n/n!, and 1/(z-1) = -\sum_{n=0}^∞ zn. Then, using Cauchy's product, and rearranging...

I am a bit confused about laurent series. I know the definitions where the coefficients are expressed as integrals. However, I am confused about how to actually find the laurent series in practice, for analytic functions. The information I can find online is just terrible, some of them do solve...

Homework Statement Determine the first three terms of the Laurent expansion in z of f(z)=\int_0^1 dx_1..dx_4 \frac{\delta(1-x_1-x_2-x_3-x_4)}{(x_1 x_2 a + x_3 x_4 b)^{2+z}},\quad a,b>0 2. The attempt at a solution I tried expanding around z = -2. f(z)=\sum_{n=-\infty}^\infty a_n...

Homework Statement Determine the coefficients c_n of the Laurent series expansion \frac{1}{(z-1)^2} = \sum_{n = -\infty}^{\infty} c_n z^n that is valid for |z| > 1. Homework Equations none The Attempt at a Solution I found expansions valid for |z|>1 and |z|<1: \sum_{n =...

Homework Statement Does the principal branch square root of z have a Laurent series expansion in the domain C-{0}? The Attempt at a Solution Well I'm not really sure what a principal branch is? I believe that there is a Laurent series expansion for z^(1/2) in C-{0} because originally our...

Let F be a field. Let c \in F. I am trying to show that if c = f^2 where f\in F[[x]], then f\in F. So I am able to get rid of the terms with negative exponent. So now I'm left with a formal power series. Anyone knows how to do this? Thanks!

1. Homework Statement For f(z) = 1/(1+z^2) a) find the taylor series centred at the origin and the radius of convergence. b)find the laurent series for the annulus centred at the origin with inner radius given by the r.o.c. from part a), and an arbitrarily large outer radius. 2...

Homework Statement I'm stuck on the following problems: I have to find the Laurent series around 1 of f(z)=\frac{z^{3}}{z^{2}-1} and the Laurent Series around 2 and on the annulus 1 < |z| < 2 of f(z)=\frac{z^{2}-2z+5}{(z-2)(z^{2}+1)} Homework Equations I am familiar with the geometric...

I have to find the Laurent series for the following cases: f(z)=\frac{1}{(z-a)(z-b)} for 0 < |a| < |b| around z=0 z=a z=\infty and on the annulus |a| < |z| < |b| I know I can do a partial fraction thingy: f(z)=-\frac{1}{a(b-a)(\frac{z}{a}-1)}+\frac{1}{b(b-a)(\frac{z}{b}-1)} which can be...

The part about Laurent series in my Complex Analysis book is somewhat vague and Wikipedia etc. didn't help me much. I am hoping someone would tell me the exact mathematical definition of a Laurent series (around a given point?) of a given function, perhaps providing an example. Also, how can...

Homework Statement How to represent function \frac{1}{e^x-x-1} in form of Laurent series around point 0 Homework Equations Laurent series f(z)=\sum^{\infty}_{n=-\infty}a_n(z-z_0)^n Here is z_0=0 The Attempt at a Solution Computer gives \frac{2}{x^2}-\frac{2}{3...

Homework Statement find the laurent series of sin(2z)/(z^3) in [z]>0 Homework Equations The Attempt at a Solution I am completely confused. I can understand some of the examples given on laurent series, like using partial fractions and then finding geometric series. Do I rewrite...

Homework Statement In attachment The Attempt at a Solution I break into partial fractions, then get stuck. Please help me in layman terms f(z) = -1/3[3(z+1)] + 4/3[z+4] Now I am stuck.

A method to get the Laurent series of a complex function is by undetermined coefficient.For example f(z)=cot(z)=cos(z)/sin(z).If we want to get the Laurent series of cot(z),we can expand cos(z) and sin(z) to Taylor series respect,then assume the series of cot(z) is a_{ - 1} z^{ - 1} + a_0 z^0...

hey there, I just studied the whole taylor and laurent series, and I think I mixed them up alittle.so here's what I know: - if we have a contor in which our f(z) is analytic completely, we can expand it in taylor series. - if we have singularities, we can expand the functions around the...