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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Verification of Laurent Series calculation

I tried to calculate the Laurent series of these functions but I have no way to verify my answers. i) $$ \begin{align} f(z)=\frac{e^{z^2 }-1}{z^4}, \mathbb{D}=\mathbb{C} \backslash \{0\} ...
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78 views

Is there a theory of “rings” with partially defined multiplication?

Consider the abelian group $R [[\mathbb{Z}^d]]$ of all formal Laurent series over a commutative ring $R$ (a typical element has the form $\sum_{v \in \mathbb{Z}^d} \lambda_v \cdot X_1^{v_1} \dots ...
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12 views

Guidance / Help with Laurent series expansion in a certain annulus

I am trying to study complex analysis and I've come across this $$ \begin{align} f(z)= \frac{1}{1+z^2} \end{align} $$ I need to determine the Laurent series expansion for the annulus ...
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34 views

Finding the Laurent series of a nontrivial function

Could anyone show me how to find the Laurent series of $$ f(z) = \frac{\operatorname{Log}(z)}{(z-1)(z-5)} $$ centered at $5$? I know how to find it for $\operatorname{Log}(z)$, using the ...
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347 views

An outrageous way to derive a Laurent series: why does this work?

I had to compute a series expansion of $1/(e^{x}-1)$ about $x=0$, and in the course of its derivation, I made a couple of manipulations that are not allowed mathematically. Still, comparing the final ...
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62 views

Laurent series for $1/(e^z-1)$

Trying to compute the first five coefficients of the Laurent series for $$\frac{1}{e^z-1}$$ centered at the point $0$. I'm not seeing a way to use the geometric series due to the exponential. Any ...
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30 views

Number of Laurent series to compute

I'm trying to calculate the Laurent series on all annuli for the function $$ f(z) = \frac{2z+i}{(z+1)^2} $$ about the point $z_0=2i$. I just want to confirm the number of Laurent series that I will ...
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49 views

Computing the Laurent series for $1/z^2$ about $z_0=1$

I'm trying to compute the Laurent series of $f(z) = 1/z^2$ about the point $z_0=1$. Looking at my notes, it appears that I need to compute a series for $|z-1| < 1$ and one for $|z-1|>1$, due to ...
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24 views

Laurent series of $f(z)=(z^2-1)\mathrm{cos}\frac{1}{z+i}$ in $z_{0}=-i$ and $Res[f(z), -i]$

This is how I've done so far: $$ f(z)=(z^2-1)\mathrm{cos}\frac{1}{z+i} \\w=z+i \;\;\;\; \Rightarrow \;\;\;\; z=w-i \\f(w)=((w-1)^2-1)\mathrm{cos}\frac{1}{w}=(w^2-2wi-2)\mathrm{cos}\frac{1}{w} ...
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89 views

Does $x^2+ x^4/(3\cdot4) + x^6/(3\cdot4\cdot5\cdot6) + \cdots$ have any compact form?

Is there any compact form for the following series $$F_1(x) = x^2+ \frac{x^4}{3\cdot4} + \frac{x^6}{3\cdot4\cdot5\cdot6} + \cdots$$ $$F_2(x) = x+ \frac{x^3}{2\cdot3} + \frac{x^5}{2\cdot3\cdot4\cdot5} ...
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44 views

Laurent series expansion of function $f(z)=\frac{1}{z^2-z}$ in powers of $z-i$ in region of convergence

I need a little help with this one. This $z-i$ confuses me. $$f(z)=\frac{1}{z^2-z}=\frac{1}{z(z-1)}=-\frac{1}{z}+\frac{1}{z-1}$$ So, singularities are $z=0$ and $z=1$. What regions do I have to ...
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58 views

Calculating Laurent Series of Complex Function

How does one alternate the Bernoulli number series expansion $$\frac x{e^x - 1}=\sum_{n=0}^{\infty}\frac{B_nx^n}{n!}$$ To calculate the Laurent Series centered at 0 in the annulus of convergence of ...
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33 views

Laurent Series expansion with |z-1|

Here's the problem: Expand $\dfrac{e^z}{z-1}$ in a Laurent series convergent in $0 < |z-1| < \infty$. And here is my attempt, I just wanna know if it's right cause it seems kinda simple. ...
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49 views

Find the Laurent Expansion of $f(z)=\frac{1}{z+i}$

Find the Laurent Expansion of $f(z)=\frac{1}{z+i}; f(z)=\frac{1}{(z-i)^2}$ and $f(z)=e^{(z-1)^-1}$ Good evening, I have been trying to solve the above exercises. However, I'm not sure if my procedure ...
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1answer
44 views

Rational Series VS Algebraic Series

I am reading a paper on combinatorics. It mentions some generating functions are rational series and others are algebraic series. I do not understand the difference, can someone help? EDIT $1$: The ...
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133 views

Why is $a_{n}(x,y)=a_{n}(y)$?

This particular question is connected (with a slight variation in the definition of $g$) to an earlier question. The link is here. The specifics are: Given that $u(x,y)$ is the solution of a PDE ($x$ ...
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26 views

A pole of function in the article of Springer

I read the article Springer, T.A. On the invariant theory of $SU_2$, Indag. Math. 42, 339-345 (1980). Author considers $\mathbb{C}$-linear map at page $340.$ If $n$ is a positive integer, then write ...
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39 views

Question regarding singularity of a complex function

Consider the function $$f(z) = {1 \over (z-i)(z+i)}$$ with a Laurent series expansion at $z_0=i$ on a domain $\;\Omega=\left\{z\in \mathbb{C}:2\lt\left|z-i\right|\right\}$ $$\begin{eqnarray}f(z)={1 ...
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22 views

Finding the convergence radius of a complex laurent series

Find the maximal ring where the following series converges: $$\sum_{n=1}^\infty\frac{3^n+2^n}{(z-5)^n}+\sum_{n=0}^{\infty}\frac{n^2}{20^n}(z-5)^{2n}$$ I think that taking the minimum between the ...
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26 views

A holomorphic function $f$ has an essential singularity in $0$ iff $\exists(z_k)_k$ s.t. $z_k\to 0$ and $|z_k^mf(z_k)|\to\infty$ for all $m$

Let $f:\mathbb{C}\setminus\left\{0\right\}\to\mathbb{C}$ be a holomorphic function $\Rightarrow$ $f$ has an essential singularity in $0$ if and only if $\forall m\in\mathbb{N}:\exists ...
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How to compute $\int_C {e^{3z}-z\over (z+1)^2z^2}$?

I am asked to compute the integral $$ \int_C {e^{3z}-z\over (z+1)^2z^2} $$ where $C$ is a circle with the center at the origin and radius ${1 \over 2}$. My approach was to separate the integral as a ...
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90 views

If $f(z):=\sum_{n=0}^\infty a_nz^{-n}$ is compact convergent, then $f$ is holomorphic

Let $\left(a_n\right)_{n\in\mathbb{N}}\subset\mathbb{C}$ such that $$f(z):=\sum_{n=0}^\infty a_nz^{-n}$$ is compact convergent on $B_r(0)\setminus\left\{0\right\}$. I want to show: $f$ is ...
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21 views

Laurent Series in a proof of the Bieberbach conjecture for $a_2$

Let $f$ be schlicht, (univalent on the unit disc with power series $f(z)= z + \sum_{n=2}^ \infty a_n z^n$ ) and let $g(z)= [f(z^{-1})]^{-1}$ I am trying to find the Laurent series of $g$ for all $z$. ...
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55 views

using geometric series rules when solving Laurent Series

when looking at Laurent Series expansions, I sometimes see something like: $$ \frac{1}{\sin z}= ...
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49 views

Finding Laurent series where given annulus is not in a singularity

I'm given a problem where I need to calculate the Laurent series of $f(z)$ inside the given annulus $$ f(z) = {1\over z^3(z-1)}; \quad 1 < |z| < 2 $$ From online resources(videos, notes) I ...
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42 views

What is the radius of convergence of this series?

Suppose we have this series: $$f(z) = \frac{1}{2z^3} + \frac{1}{12z} - \frac{z}{240}.$$ What is the radius of convergence?
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Laurent series of $\frac{e^{iz}}{z^2+p^2}$, $ p>0$.

I need help finding the main part of the laurent series of $f(z)=\frac{e^{iz}}{z^2+p^2}$ in $ip,-ip$ since these are the two poles of $f$. Due to the orders of the poles are 1 I just have to find ...
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3answers
65 views

Laurent series and residue of $f(x)=\frac{1}{1+e^z}$

I am having trouble trying to expand this function using Laurent series, and finding the residue$$f(x)=\frac{1}{1+e^z}$$ If I replace $e^z$ with its series I get ...
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What is the Laurent series of $ \exp \! \left( - \dfrac{1}{z} \right) $?

I’m thinking that I could simply let $ x = - \dfrac{1}{z} $ in the Maclaurin series for $ e^{x} $: $$ 1 - x + \frac{x^{2}}{2!} - \frac{x^{3}}{3!} + \cdots = 1 - \frac{1}{z} + \frac{1}{2! z^{2}} - ...
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1answer
57 views

Find the expansion of $\frac {1}{z^2+3}$ valid for $|z-1|<2$

Find the expansion of $$\frac {1}{z^2+3}$$ valid for $|z-1|<2$. Should I start with considering $z-1=u$ ? It gets messy that way. I tried reorganizing the denominator of $f(z)$ but that didn't ...
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49 views

Expand $ f(z)=\frac {e^{2z}}{(z-1)^3}$ about $z=1$ as a Laurent series

Expand $$f(z)=\frac {e^{2z}}{(z-1)^3}$$ about $z=1$ as a Laurent;s series. Also indcate the region of convergence. My attempt : Let the given complex function be analytic in an annulus ...
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1answer
39 views

Largest Open set where Laurent Series Converges

I need to know the largest open set on which Laurent series of $f(z)={z^2-\pi^2\over \sin z}$ converges. $f(z)={(z-\pi)(z+\pi)\over \sin z}$ clearly as $z\to \pm\pi,f(z)\to \pm2\pi$, so $f(z)$ extend ...
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75 views

Quick question on infinite complex integral

I'm supposed to find: $$ \int_0^{\infty} \frac{(\ln x)^2}{1+x^2} dx $$ I start of by finding: $$ I_1 = \oint \frac{(\ln z)^2}{1+z^2} dz $$ $$ = \oint \frac{(\ln z)^2}{(z+i)(z-i)} dz $$ Now I ...
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Prove that periodic analytic function can be written as $\sum_{-\infty}^{\infty} c_n e^{2\pi inz}$

This question involves the following homework problem: PROBLEM Suppose $f$ is analytic in the upper half plane and periodic of period 1. Show that $f$ has an extension of the form ...
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Classification of Singularities and Residues

I'm kind of stumped on a question here. I've been asked to determine and classify the singularities of; $$f(z) = \frac{z^3}{(1+z)^3}$$ To me, it's pretty obvious that a singularity will occur when ...
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integration of a multiple Laurent series

let be an multiple integral given by $$ \int_{0}^{\infty}dx_{1}\dots\int_{0}^{\infty}dx_{n}F(x_{1} ,x_{2},...x_{n}) $$ i have a question can i solve this integral equation by exapnding the integrand ...
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Laurent-Series Expansion and Residues

I was asked to find a Laurent-series expansion for $f(z) = z^{-1} \sinh(z^{-1})$ about the point 0, and to classify the singularity at 0. Now, that wasn't too bad - with some manipulation of the ...
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65 views

Residue of this function for $z_0=0$

I have this function $$\frac{\sin (2z)-2z}{(1-\cos z)^2}$$ I want to find its residue around $z_0=0$, however I've been battling it for hours but I get nowhere. I've tried finding its Laurent series, ...
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Not so clear singular points

Find the singular points and determine the type: $g(z)=\frac{z^2-3z+2}{z^2-2z+1},z_0=1$ $f(z)=\frac{1}{1-\sin(z)}$ For $g$, we have that $$ ...
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Is it correct to say that this series diverge in the given annulus?

Let's consider the annulus $1\le|z|\le2$ and $$f(z)=\frac{2z+3}{z^2+3z+2}$$ Find its Laurent series. First I simplify the expresion: ...
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26 views

Find Laurent's series of these two functions around $z_o$

Find the Laurent series of $f(z)=\frac{z}{(z+1)^2}$ around $z_o=-1$, and $g(z)=z\exp(\frac1{z+i})$ around $z_o=-i$. For $f$, what they're asking is to find the series in $0<|z+1|$. On the ...
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Is there any example of usage for a vector space over the field of formal Laurent series?

The formal Laurent series over a field is a field. Is there any example where vector spaces over that field occur naturally?
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50 views

Find the Laurent series of $\sin z/z^2$ using Laurent's theorem

I have the function $f(z)=\frac{\sin z}{z^2}$, wich is analytic over $\Bbb C\setminus\{0\}$, I want to find the Laurent series of $f$ valid for $0<|z|<R\le\infty$. Using Laurent's theorem we ...
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1answer
39 views

Laurent series for $\frac1{z^2+1}$

I have this problem: Find the Laurent series around $z=0$, for $\dfrac{10}{(z+2)(z^2+1)}$ in the region $1<|z|<2$. I did partial fractions and found this: $\dfrac{2}{z+2}-\dfrac{2z-4}{z^2+1}$, ...
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59 views

Why is the series expansion of $\sin(x)$ at $x=n\pi$ different to the expansion of $\sin(2x)$ at $x=\frac{n\pi}{2}?$

I'm trying to do a series expansion of $\sin(2x)$ about the point $x=\frac{n\pi}2$ where is an integer. I thought that the expansion would be the same as for $\sin(x)$ about the point $x=n\pi$ but ...
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39 views

Find the Laurent Series for $|z|>3$

Hi this is my question, $$f(z)=\frac{2}{(z-2)(3-z)}$$ Find the Laurent Series for $|z|>3$ of $f(z)$ I have split it into partial fractions and have ended up with $$ 2\sum_{n=0}^\infty ...
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1answer
37 views

Find the Laurent series expansion in powers of z

Find the Laurent series expansion in powers of $z$ of $$f(z)=\frac{e^{2z}} {z}$$ valid in the region $|z|>$0. Any help appriciated. Thanks
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33 views

Power series expansion of $e^{\frac{c}{2}(z-1/z)}$

Show that \begin{equation} e^{\frac{c}{2}(z-1/z)}=\sum_{n=-\infty}^{\infty}a_nz^n \end{equation} where \begin{equation} a_n:=\frac{1}{2\pi}\int_0^{2\pi}\cos(n\theta-c\sin(\theta))d\theta ...
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1answer
75 views

Laurent expansion of $\frac{1}{1+z^2}$ in $A:= \{z \in \mathbb{C} : | z - z_0 | \gt |z_0 + i|, Im(z_0) \gt 0\}$

I need to find the Laurent expansion of $f(z) := \dfrac{1}{1+z^2}$ in the set $A:= \{z \in \mathbb{C} : | z - z_0 | \gt |z_0 + i|, Im(z_0) \gt 0\}$. I've drawn a picture of this: I know that if $r ...
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1answer
20 views

Laurent series of quotient

If I have two functions $f,g$ that are holomorphic around a point $z_0 \in \mathbb{C}$. Assume the Laurent series are known and both $f$ and $g$ have a finite principal part. $$f(z) = ...