The Laurent-Series is a generalisation of the power series which allows negative indices and is essential for investigating the behaviour of functions near poles.

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Riemann zeta, why are the residues either zero or one?

One more question, probably equally simple to answer but I don't know how this is true either: Why is the residue of Riemann zeta zero - trivial or non-trivial: $$\text{residue}\left(\zeta ...
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Pole of Riemann zeta and Riemann zeta zeros, prove this relation.

Prove this relation: $$\displaystyle \lim_{s\to 1} \, \left(\zeta (s)-\frac{\zeta '(s-1+\rho _n)}{\zeta \left(s-1+\rho _n\right)}\right)=\gamma -\frac{\zeta ''(\rho _n)}{2 \zeta '(\rho ...
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49 views

Complex Taylor and Laurent expansions

Let $f(z):=\dfrac{1}{2-z-z^2}, z\in\mathbb{C}\setminus\left\{ {1, -2}\right\}$. i) Express $f$ in the form $\dfrac{A}{1-z}+\dfrac{B}{2+z}$. [Answer to this is $\dfrac{1/3}{1-z}+\dfrac{1/3}{2+z}$]. ...
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Evaluate an integral $\int_{\gamma } z^{n}e^{\frac{1}{z}}dz$ maybe laurent series or taylor expansion?

Please help me with this one guys, I am stuck like a truck trying to get out of thick mud. Evaluate: $\int_{\gamma } z^{n}e^{\frac{1}{z}}dz$ $\gamma$ is the circle f radius 1 centered at 0 and ...
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1answer
44 views

Laurents Series Expansion Complex Analysis

So here is the problem, I am having a lot of trouble with laurents expansions and if you guys even know any sources where I can learn these really well and very simply then that would be a great help. ...
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30 views

Find the Laurent series of the function and give the residue

I have $f(z)={z^2\over {z^2-1}}$. I want to find the Laurent series of $f$ and the residue at the point $z_0=1$. Can I say that $f$ has a pole of order 2 at $z=1?$ Or is that only used when the ...
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42 views

Integral of Difference of Logs

I get the expansion of $h$ to be $$ h(z) = {1 \over z } \sum_{r=1}^{\infty}{1 \over r}{(-{\alpha \over z}})^r $$ $$ \Rightarrow h(z) = \sum_{r=-2}^{-\infty}{{(-\alpha)^{r+1} \over -(r+1)} z^{r}} $$ ...
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What is the Taylor series of $\frac{1}{\sin(z)}$ about $z_0 = 1$?

This was a exam question so I know it cannot take too long to write out the proof. Only I cannot see an answer. I would imagine you write $\sin(z) = \sin(1+(z-1)) = \sin(1)\cos(z-1) + ...
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92 views

Calculating residue of $z\sin{ \frac {z+1}{z-1}}$

Let $f=z\sin{ \frac {z+1}{z-1} }$. Calculate the residue of $f$ in $z=1$. I think $f$ has an essential singularity at $z=1$ so the only way I can proceed is with Laurent series. I've defined ...
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49 views

Laurent Series and Taylor Expansion of $ 1 / (e^z - 1) $

Could someone please assist me with the second part of the second paragraph, from "By expanding $f_1$..."? I am not convinced that my expansion for $f_1$ is right - I used the standard binomial, ...
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1answer
31 views

Expand $\frac{z}{(z^2 + 1)(z^2 + 4)}$ for $1 < | z | < 2$ in Laurent series

Please help. Expand $\frac{z}{(z^2 + 1)(z^2 + 4)}$ for $1 < | z | < 2$ in Laurent series.
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1answer
67 views

Laurent series expansion, principal part

I need to find the principal part of the Laurent series for $f(z) = \frac{e^{2z}}{1-\cos(z)}$, around $z = 0$. Also, I have to use the undetermined coefficient method. I don't know how to proceed. ...
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1answer
37 views

Laurent series expansion help

Fint the Laurent series for: $f(z)=\frac{3z-3}{2z^2-5z+2}$ in the annulus $\frac{1}{2}\lt|z-1|\lt1$ I know it requires a bit of fiddling then using the formula for geometric expansion, but I'm ...
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1answer
29 views

Laurent Series for this function

$f(z) = \frac{1}{(z^2 + 1)^2}$ in the domain $\{z: 0<|z-i|<2\}$ but I'm struggling to get to the result, thanks in advance for your time
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1answer
84 views

About asymptotic behaviour of a divergent integral.

I have the function $f(x) = x \tanh(\pi x) \log (x^2 +a^2)$ where $a$ is some positive real number. For the logarithm I am assuming a branch-cut along the positive imaginary axis starting at $x = ia$. ...
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1answer
96 views

How to estimate the growth of a “savage” function near 1?

Say I have a function which exists within the unit disk, say $$f(x)=a_0+a_1x+a_2x^2+...$$ If we know sufficient information about the coefficients, say we know the growth rate of ...
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1answer
24 views

laurent series of a function defined by an integral

let be the 2 functions defined by an integral a) $$ F(x)= \int_{0}^{x}g(t) dt $$ b) $$ H(x)= \int_{0}^{\infty} dt \frac{g(t)}{1+tx} $$ assuming that i only can evaluate $ F(x)$ and $ H(x)$ by ...
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24 views

Laurent Series Regions

Suppose that I'm trying to construct Laurent series for the following complex function: $$f(z)=(z^2+4)^{\frac{1}{3}}$$ Since I have a 1/3 exponent, this means that my branch points must extend to ...
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52 views

Finding coefficients of laurent series for $\frac 1{1-\cos z}$ about zero

I know I'm wrong, but I fail to see why I'm wrong. My goal is to try and find the terms for the Laurent series of $f(z)=\frac{1}{1-\cos(z)}$ but I'm surely off. $$\begin{align} f(z)&= ...
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29 views

Laurent Series Region

So, I'm trying to find the regions in which the function may be represented as a Laurent series (expanded about the origin). Then I want to find those expansions: $$f(z)=(z^2+1)^{-1/2}$$ Now, since ...
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42 views

Odd Laurent Series

So, I've been studying Laurent series, and I'm fine with series such as $ \frac {1}{(z-1)(z+1)} $ for example. For these, we can just use partial fraction decomposition and then geometric series. ...
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Laurent Series of Riemann Zeta Function

How do I go about finding the Laurent series of the Riemann zeta function about $z=1$?
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61 views

Integral representation of the Bessel function (J)

Laurent series expansion of the generator function gives, $g(z,t) = e^{z/2(t - 1/t ) } = \sum J_n (z) t^n. $ The term $t^n$ suggests that this expansion is performed around the origin, so we have, ...
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Laurent Series, Cauchy, Pole Order

http://gyazo.com/8ef04b854bc3bbfb6b55a9af45e51fdc.png Since f(z) not differentiable at 0, isolated singularity at z = 0. By expanding the Laurent series and looking at the first term, I got a pole of ...
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1answer
36 views

Laurent Series / Residue Theorem

I'm having trouble on computing $\int_\gamma \frac{dz}{(z^2-4)(z-2)}$, where $\gamma$ is the positively oriented circle centered at 2 of radius 1. Any help on this will be very appreciated.
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27 views

Spectral Radius given by Neumann Series

Good afternoon everybody. So far it is clear that outside the spectral "disk" the resolvent is given by the Neumann Series. But why does it tell us that there is a point on the spectral "circle" where ...
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81 views

Trouble with determining principal part of function at a pole

In Fischer's $\textit{A Course in Complex Analysis}$ I am encountering some difficulty in explicitly calculating the principal part of a function at a pole. The function is $f(z)= \frac{1}{z - \sin ...
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2answers
84 views

About the integral $\int_{0}^{+\infty}\frac{\sin(ax)\,dx}{x(x^2+1)}$

I need to prove the following identity: $$\forall a>0,\qquad\int \limits_{-\infty}^{+\infty}\frac{\sin(ax)\,dx}{x(x^2+1)}=\pi(1-e^{-a}).$$ I think it can be proven using Laurent series. I tried ...
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103 views

Calculate Laurent series for $1/ \sin(z)$

How can calculate Laurent series for $$f(z)=1/ \sin(z) $$ ?? I searched for it and found only the final result, is there a simple way to explain it ?
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40 views

What is the Laurent series of the complex absolute value?

What is the Laurent series of the function $f(z) = |z|$? It seems to be ill defined at $z=0$. Are there any other expansion techniques applicable for this function at $z=0$?
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23 views

Laurent series and residue.

Why is $\sum_{n=0}^\infty (1/w)^{2n+1} = \sum_{m=-\infty}^0 w^{2m+1}$, using the identity ($m=-n$)? Why shouldn't the exponent be equal to $2m-1$? Also, why is the following correct - ...
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1answer
63 views

Laurent Series: sec(z) centered at origin

i'm having a little trouble in finding the laurent series for sec z, centered at the origin. I'll be appreciated for any help given. Thanks.
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Residue of a a complex quotient

I have the following Laurent expansion corresponding to the function: $$f(z)=\frac{z+2}{z^2-4iz-3}$$ $$f(z)=\left(-1+\frac{1}{2}i\right)\sum_{n=1}^\infty ...
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1answer
37 views

Ratio of coefficients for Laurent series expansions [duplicate]

Let $f$ be analytic in the disk $D(0,2)$ except for a pole of order $1$ at $z=1$, and let $$f(z)=\sum_{k=0}^\infty a_k z^k$$ be the series expansion for $f$ in the disk $D(0,1)$. Prove that ...
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1answer
46 views

Laurent expansion of digamma function around $x=0$

I want to check the validity for such a method Define the digamma function as $$\psi_0(x)=\frac{d}{dx}\left( \log \Gamma(x)\right)$$ $$\tag{1}\psi_0(x)=\frac{\Gamma'(x)}{\Gamma(x)}$$ It has the ...
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1answer
122 views

Laurent series for $\frac{1}{z^2\sinh(z)}$

Calculate the 3 first terms of the Laurent series for $f(z)=\displaystyle\frac{1}{z^2\sinh(z)}$ where $0<|z|<R$ and calculate the highest possible value for $R$. I've figured out I can do the ...
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1answer
60 views

Laurent series for $\frac{e^z}{z^2}$

Calculate the Laurent series of $\displaystyle\frac{e^z}{z^2}$ when $0<|z|<\infty$. I know how the terms $a_n$ and $b_n$ of the Laurent series are defined in terms of a closed integral, but I ...
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1answer
53 views

Laurent series for $\frac{z}{z+1}$ when $1<|z|<\infty$

Calculate the Laurent series for $\displaystyle\frac{z}{z+1}$ when $1<|z|<\infty$. There is really no singularity here, right? Can I just use a Taylor series, or what should I do?
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43 views

Laurent Seies and Res

Prove that for any Laurent series f(t) one has "Res(f') = 0"? I know for a Laurent series of a complex function f is a representation of that function as a power series which includes terms of ...
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60 views

Laurent series expansion of sine

The problem is : find the expansion of $ \frac {z}{\sin z} $ in the ring $\pi < abs (z) < 2\pi $, if the expansion in $abs (z)<\pi $ is known. How to solve such problems?
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Laurent Series Difficulty

Hello all at StackExchange, I'm having some trouble understanding computing the Laurent series for different domains. Here's my approach to finding the Laurent series for $\dfrac{3}{(z+1)(z-2)}$ for ...
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1answer
39 views

Laurent series of $e^z/(z^3+z)$

Find the first four terms of the Laurent series of $$\frac{e^z}{z^3+z}$$ in the punctured disk $0<|z|<1$. We can apply the formula to get $$f(z)=\sum_{k=0}^\infty a_kz^k$$ where ...
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1answer
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Conceptual question regarding laurent series

So I have a physics background, not math, but i'm trying to understand at least conceptually WHY laurent series are computed in the way they are. For example, take the series expansion of ...
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1answer
26 views

Gived y[n], a discret signal, using Z transform, get the general expression.

i hope you can guide me in this mess i have. Gived this signal $y[n+2]-y[n+1]-30y[n]=(1/5)^n+1; \qquad (n \in \Bbb N, n>= 2) $ $y[0]=0,$ $y[1]=1.$ get the general y[n] expression. Applying Z ...
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1answer
52 views

Laurent series, radii of convergence.

I'm working on the following exercise: Prove that a Laurent series \begin{align*} \sum_{n = -\infty}^\infty a_n(z-z_0)^n = \sum_{n = 0}^\infty a_n(z-z_0)^n + \sum_{n = 1}^\infty ...
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2answers
58 views

Calculate $\int_\Gamma \frac{f'(z)z}{f(z)}\, \operatorname dz$

Calculate usign the formula for zeros and poles, for a meromorphic function $f$ the following: $$\int_\Gamma \frac{f'(z)z}{f(z)}\, \operatorname dz$$ Where $\Gamma$ is simple and closed. I tried ...
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1answer
48 views

$\varphi$ the Laurent series near pole $z_0$, how prove that $f-\varphi$ is analytic near $z_0$?

I'm reading the proof that a meromorphic function on the Riemann sphere must be a rational function, and I think I need to understand Laurent series better. The idea of the proof I'm reading is that ...
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63 views

Need Hint on Laurent series

I stuck at part (a) after 1 have done the partial fraction, can anyone give me some hints? for part (b) I express $$f_1(z)=(-1)^n-1(n-1)!f_n(z)$$ What I suppose to do next?
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1answer
86 views

Laurent Series of $(z^2-1)^{-2}$

I have problems to determine the Laurent series of the function $(z^2-1)^{-2}$ in the regions: $$0<|z-1|<2$$ and $$|z+1|>2$$. My idea was as follows: ...
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1answer
37 views

Coefficients for expansion is unique

Suppose $$f(z)=\sum_{n=-\infty}^\infty a_nz^n$$ for all $1<|z|<2$. Show that the coefficients $\{a_n\}$ are unique. I want to use the fact that the Laurent series of any function $f$ is ...