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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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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+50

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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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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35 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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19 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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21 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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87 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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15 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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54 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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43 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
61 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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48 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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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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34 views

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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1answer
63 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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33 views

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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31 views

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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23 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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45 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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37 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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58 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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37 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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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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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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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) = ...
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59 views

Complex Analysis: Isolated Singularities, Poles, and Residues

I was given the following question. Show that the isolated singularities of the function $f(z) = \frac{z}{z^4+4}$ are poles. Determine the order of each pole and find the corresponding ...
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Complex Analysis: Laurent Series Expansion

I am given the following. Let $f(z) = \frac{e^z-e^{-z}}{z^4}$. Give a Laurent Series for $f(z)$ in terms of powers of $z$. For which values of $z$ does the Laurent series converge? Justify ...
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Calculate the value of the integral of a series

let $$P(r,\varphi):= \dfrac{1}{2\pi} \sum_{n \in \mathbb{Z}} r^{|n|}e^{in\varphi} $$ with $\varphi \in \mathbb{R}$ and $ 0< r <1$. Prove that $$\int_{0}^{2\pi}P(r,\varphi)d\varphi =1$$ My ...
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60 views

Finding Laurent expansion of $\frac{2}{(2z-3)(z-5)}$.

Find the Laurent expansion of $\frac{2}{(2z-3)(z-5)}$ that is valid in the annulus $\frac{7}{2} < |z-5| < \infty$. There is a lot to write out, but here is the basics of the work. We can leave ...
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41 views

Laurent Expansion partial fractions

I have a function: $\frac{1+2z}{z^3 + z^2}$ for $0 < |z| < 1$ (about $z=0$) I need to find the Laurent expansion of this function. However, I'm a bit confused how to find the partial fractions ...
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46 views

Laurent Series of $f(z) = \frac{1}{e^z - 1}$

The Laurent Series of $f(z)$ centred at $0$ can be written as, $$f(z) = \frac{1}{z} - \frac{1}{2} + \frac{z}{12} - \frac{z^3}{720} + \cdots$$ So we see that $f(z)$ has a simple pole at $0$. Can we ...
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25 views

Calculating the quotient of 2 Laurent series

Let $D$ be an open subset of $\mathbb{C}$ and $f,g : D \rightarrow \mathbb{C}$ be meromorphic, both having a pole at $z_0$. If I know the Laurent series of $f$ and $g$ around $z_0$, how to calculate ...
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30 views

Fourier Series to Laurent Series

Given a periodic function $f(\sigma)$ with period $T$, one can compute its Fourier series, $$f(\sigma)=\sum_{n\in\mathbb{Z}} c_n e^{i \omega n\sigma}$$ where $\omega=2\pi/T$ and the coefficients of ...
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128 views

Computing the Laurent series of $e^{z+\frac{1}{z}}$

I'm really stuck on this, and I have no idea how to start. Writing it at $f(z)=e^z e^{\frac{1}{z}}$ and their expansions didn't really give any insight. I am aware it is possible to multiply the ...
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36 views

$\sin(1+\frac{1}{z-1})$ expanded in powers of $z-1$

The whole problem: Obtain the following Laurent expansions. State the first four nonzero terms. State explicitly the $n$th term in the series, and state the largest possible annular domain in which ...
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90 views

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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1answer
66 views

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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75 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 ...