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Questions tagged [laurent-series]

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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Finding the Laurent Series around a given point

While studying I got this exercise: Find the Laurent Series expansion valid for $0 < |z - i| < \sqrt2$ for the following function:$$f(z) = \frac{1}{(z-i)^8(z+1)}$$ So I have to get a series ...
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Residue and Laurent Series, is this valid?

something with the Laurent series is confusing me, first I'll give a background of what I think I know. If $z_0$ is an isolated singularity of a function $f$ we can find the $Res(f, z_0)$ by finding ...
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Calculation of Laurent series and decomposition in partial fractions

So I was asked to find the Laurent series of the expression around $z=2$: $$f(z)= \frac{z+3}{(z-2)^3}$$ I am aware that this can be decomposed as $$5\, \left( z-2 \right) ^{-3}+ \left( z-2 \right) ^{-...
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Advice on using this alternative method of finding the Laurent expansion of $\tfrac{1}{z^2(z-1)}$

Say we want to calculate the Laurent series of $\tfrac{1}{z^2(z-1)}$ about $z_0=0.$ Now I know that one way to do it is to say that $f(z)=\tfrac{1}{z^2}(\tfrac{1}{z-1})$ and appy the geometric series ...
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Expand $\operatorname{Log}\frac{z^2}{z^2-1}$ into Laurent series for $|z|>1$ [closed]

Expand $\operatorname{Log}\frac{z^2}{z^2-1}$ into Laurent series for $|z|>1$ I have simplified the expression to: $$ \operatorname{Log}\frac{1}{1-\frac{1}{z^2}} $$ but I am not sure what to do ...
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Decide which term represents the negative exponent on Laurent series.

This might be a stupid question, but here we go. I cannot understand the logic behind each part of a Laurent series, for $n\geq0$ and $n<0$. Here is an example: Find the Laurent series in the ...
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Laurent series of $\frac{z}{\sin({\frac{\pi}{z+1}})}$ in the roots of the denominator

I'd like to compute the first few terms of the Laurent series of $\frac{z}{\sin(\frac{\pi}{z+1})}$ at $z=\frac{1}{k}-1, k\in\mathbb{Z}$. I assume I know the espansion of $\frac{1}{\sin{z}}$ in $z=0$, ...
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Residue of $f(z)=\frac{z}{\sin{\left(\frac{\pi}{z+1}\right)}}$ in all isolated singularities

I have this complex function: $$f(z)=\frac{z}{\sin\left(\frac{\pi}{z+1}\right)}$$ I'd like to compute residues in all isolate singularities. If I'm not mistaken $f$ has poles in $z=\frac{1}{k}-1$ and ...
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Laurent serie of $ \frac { \cos z}{ \sin z + \sinh z - 2z}$

I'm working on an example given in my book of complex analysis: $$ \frac { \cos z}{ \sin z + \sinh z - 2z}$$ but I can't figure out how he finded the residue in 0. The few steps he is showing make ...
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Question about integrating a Laurent series

I want to expand the function $ f(z) = \frac {z-1}{z^2 -2z -3} $ in $ 0 < |z+1| < 4$ Then I want to use the result to evaluate this integral $ \int_C \frac {z-1}{(z+1)(z-3)} dz $ in $ C : |z+...
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Why is $z_0=0$ not an essential singularity of $\tan(1/z)$ What is this singularity?

Since for any $\epsilon>0~\exists |z|=|\frac{2}{(2k+1)\pi}|<\epsilon$ for some $k$ large enough such that $\tan(1/z)=\pm\infty$ there cannot be a Laurent series at $0$. Does there need to be a ...
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Calculating Laurent expansions using geometric series in regions defined by inequalities

So I am a bit confused about Laurent series and their relation to geometric series. I will give an example but my doubts are more general. The function: $$f(z) = {\frac {{{\rm e}^{z}}}{ \left( z-1 \...
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Calculating Laurent expansion for $\frac{1}{1-z^2}$ [closed]

I don't have any idea where should I start for calculating Laurent expansion for the following function : $$\frac{1}{1-z^2}$$ The thing that I've got in my notes is finding a form like the power ...
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Find the residue, state the nature of the singularity, find the constant term in $1/\sin(ze^z)$ at $z=0$

Find the residue, state the nature of the singularity, find the constant term in series $1/\sin(ze^z)$ at $z=0$. We can rewrite the function $\frac{1}{\sin(ze^z)}$ as $\frac{ze^z}{\sin(ze^z)}\cdot\...
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Circumventing Potentially Illegal Substitutions in Generating Functions

This question comes from trying to solve the following problem from Probability: An Introduction by Grimmett and Welsh. (b) In a two-dimensional random walk, a particle can be at any of the points $...
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Finding Laurent Series using Binomial Theorem - HOW?

I'm working on a fairly simple question asking to work out the necessary branch cut(s) for the function $f(z)=(z^2+1)^{1/2}$. I am comfortable doing this and the rigour required to explained why I ...
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Finding the Laurent series (complex numbers)

I have $$ f(z)={\frac{1}{z(1-z)}} $$ Need to find the Laurent series around $z=0, z=1, z=\infty$. I did $$ {\frac{1}{z(1-z)}} = {\frac{A}{z}}+{\frac{B}{1-z}} $$ and found $A=1, B=1$. Therefore we get $...
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Find complex residue and Laurent series expansion

$$ w = \sin(z) * \sin(\frac{1}{\:z}) $$ special point is $$ z_0 = 0 $$ $$ \lim _{z\to 0}\left(\sin\left(z\right)\cdot \:\sin\:\left(\frac{1}{z}\right)\right) $$ isn't exist, next I have to decompose ...
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Finite order polynomial convergence to fraction

Happy holidays! So suppose that I have a finite order polynomial expressed as: $$f(x)=\sum_{i=0}^n a_ix^i$$ where $a_i \in \mathbb{R}$. The roots of $f(x)$ can be real or complex. Can $f(x)$ be ...
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Residue of a product of series

I need to find the residue of $f(z)=\frac{e^{\frac{1}{z}}}{z-1}$ in $z=0$. To do this, I proceeded to find the Laurent series of $f$ which is: $\sum_{n=0}^{\infty} \frac{z^{-n}}{n!}\sum_{k=0}^{\...
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Classify the singularities of $f'(z)$ and its residue

I want to know which singularity does $f'(z_o)$ have if $f(z)$ has a singularity in zo. I know that if $f(z)$ has an essential singularity in $z_o$, then its Laurent series has infinite negative ...
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Principal part of function at a pole

I have a function $\dfrac{e^zz}{z^2-1}$. It has isolated singularities $z=\pm 1$. To find the principal part at $z=1$, I am trying to find a Laurent series expansion around $z=1$. I have the following ...
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Laurent Series in powers of $z$ and $\frac{1}{z}$

I'm working on a few problems from my textbook and have a bit of trouble figuring out a few things. In a particular case, suppose $R$ is a rational function all of whose poles in the plane have ...
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Series representation for a specific range

I am wondering if there is a valid series representation using: $f(z) = \sum_{k=-\infty}^{\infty} a_k(z-z_0)^k$ for $r<|z−z_0|<R$ Why is this not possible?
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Laurent expansion of $\frac{1}{z^2+i}$ at $z=i$

I want to find the Laurent series of $1\over(z^2+i)$at $z=i$. How do I start and go on from there? Do I start with either of these? $$1 \over{(z+i\sqrt i)(z-i\sqrt i)}$$ $$\frac{1}{i} \frac{1}{1+(z^...
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Order of poles on a function

How can I determine what the order of the pole on the following function is: $$\ f(z)= \frac{e^{bz}}{z\sinh(az)}$$ From the Laurent series, I found that the residue would be b/a or -b/a, however, I ...
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Using the residue theorem to evaluate an integral

$$I = \int_0^{\infty} \frac {dx}{x^6+1}$$ My thinking is that I can use the property of even functions to integrate across the whole domain from negative infinity to positive infinity. Can the ...
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Laurent expansion of the tangent function about a given point

I am given the following question : Show that if $\tan z$ is expanded into a Laurent series about $z = \pi/2$, (a) the principal part is $-1/(z - \pi/2)$, (b) the series converges for $...
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Laurent expansion of the given function

I have the following question : Find a Laurent expansion of $f(z) = \dfrac{z}{(z^2 + 1)}$, valid for $|z-3| > 2$. I have learnt finding the Laurent expansion of functions where the denominator is ...
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What is the definition of $\sum_{n = - \infty}^{\infty} a_n$

I know that in Complex Analysis we use sums of the form $\displaystyle\sum_{n = - \infty}^{\infty} a_n$ What is the actual meaning of this symbol? I expect that under some nice enough conditions, it ...
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Laurent Series expansion about the point $z_0 = i$ of $\frac{z}{z^2+1}$

I am trying to construct the Laurent series expansion of $f(z) = \frac{z}{z^2+1}$ about $z_0 = i$ in the region $\{z \in \mathbb{C}: 0 < |z - i| < 2\}$ but I am stuck. We can re-write $f(z) = \...
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Help understanding the Laurent series

I just wanna see if I have some of the fundamentals nailed in terms of understanding. Say we have some complex valued function defined as $$f(z)=\tfrac{1}{z(z-1)}$$ and we want to evaluate it's ...
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Laurent Series Expansion of Exponential function.

Find the Laurent series of the function $f(z) = e^{\frac{\lambda}{2}\big(z-\frac{1}{z}\big)}$ as $\sum_{n=-\infty}^{\infty} C_nz^n$ for $0<|z|<\infty$ where $$C_n = \frac{1}{\pi}\int_0^{\pi} \...
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Finding laurent series expansion at Infinity

Find the laurent series expansion at $\infty$ of the follwoing function: $\dfrac{1}{z^2-8z+25}$ Consider $\dfrac{1}{z^2-8z+25}$,Replace $z=\frac{1}{w}$,we get $\dfrac{1}{z^2-8z+25}=\dfrac{w^2}{...
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Calculating the residue of $\frac{1}{z^2 \sin z}$ at $z = 0$

I am having some difficulty calculating the residue of $\frac{1}{z^2 \sin z}$ at $z = 0$. From what I can tell, we are dealing with an essential singularity here and so the problem becomes that of ...
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Is f the zero function?

I am told to find the error in the following argument: let $f(z)= \cdots + \frac{1}{z^2} + \frac{1}{z} + 1 + z + z^2 + \cdots$ Note that $z + z^2 + \cdots = \frac{z}{1-z}$ and $1 + \frac{1}{z} + \...
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If $f : D(0;1) \setminus \{ 0 \}$ holomorphic such that $f(1/n) = 0$ then either $f=0$ or $f$ has an essential singularity at $0$

I'm trying to solve this question: Let $f$ be a holomorphic function on $D(0; 1)\setminus\{0\}$ with the property that $f(1/n) = 0$ for every positive integer $n$. Show that $f$ is either ...
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Laurent series of $ \frac{z-12}{z^2 + z - 6}$ for $|z-1|>4$

How do you find the Laurent series for $f(z) = \dfrac{z-12}{z^2 + z - 6}$ valid for $|z-1|>4$? I know that $f(z) = \dfrac{z-12}{z^2 + z - 6} = \dfrac{-2}{z-2} + \dfrac{3}{z+3}$ It is easy for ...
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Verification of Laurent series for the function $f(z)=\frac{2}{(z+2)^2}-\frac{5}{z-4}$

Suppose that $$f(z)=\frac{2}{(z+2)^2}-\frac{5}{z-4}.$$ Find the Laurent series for $f$ in powers of $z-2$ that converges when $z=1$. This is how I approached the problem: \begin{align} f(z)&=-2\ \...
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Residue of order 3 -

Find the Laurent Series for the function \begin{align} f(z) = \frac{1}{(z^2+4)^3} \end{align} about the isolated singular pole $z = 2i$. What is the pole order? What is the residue at the ...
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Calculate the Laurent series centered at i on an annulus

Hi, I'm trying to solve part b of this question. I got the laurent series centered at 0 (part a) by using the geometric series and partial fractions to manipulate the expression to match the boundary ...
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General strategy in finding Laurent Series

I'm taking my first course in complex analysis and I understand everything but how to find Laurent series. Other than seeing worked examples I can never figure them out myself. One of the textbook ...
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Laurent series $f(z)=\frac{\sqrt{z}}{z+i}$

I came across to calculate an integral for which I had to find the Laurent series of $f(z)=\dfrac{\sqrt{z}}{z+i}$. I see that at $z=-i$ the function has a simple pole and the Laurent series I got is ...
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Laurent series advantage

Laurent for a function is a generalization of the Taylor series. With a Laurent series, however, the powers can be negative. An advantage of the Laurent series over the Taylor series is expanding ...
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Double checking a coefficient for a Laurent series, the series for $f(z) = \frac{z^2 + 1}{(z - i)^2}dz$

So we seek the coefficient $a_{-1}$ of the Laurent series for $f(z) = \frac{z^2 + 1}{(z - i)^2}dz$, where Laurent series is denoted by $$f(z) = \sum_{n = -\infty}^\infty a_n(z - z_0)^n$$ when the ...
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Is taylor's series always a special case of a laurent series without negative coefficents?

Is taylor's series always a special case of a laurent series without negative coefficents? In order words, If I were to draw a venn diagram of the set of all Laurent series, would taylor's series fit ...
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What is the Laurent series of $f(z) = \frac{1}{z^2 (z-3)^2}$ around $z = 3$

I tried $$\frac{1}{z^2(z-3)^2} = \frac{1}{((z-3)^2 - 6(z-3) + 9)(z-3)^2 }$$ but this doesn't lead somewhere. Am I supposed to calculate the coefficients using $$a_n = \frac{1}{2\pi i }\oint_\...
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44 views

Finding the Laurent series of $f(z) = e^{1/z}$ about $z_0 = 0$

Specifically, I want to do this without invoking the Taylor/power series definitions for $e^z$. I know they exist and accept they're valid methods, but I want to show them from the definition of ...
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Help calculating Residue at an essential singularity

Can u help me calculate the residue of the function $f(z)= (z^2/(1+9z^2))(e^{i/z}-1)$ at 0? Ive tried using the Laurent series but its seems very confusing to figure it out, and ive tried to use the ...
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Find the order of the pole of $f(z)=\frac{e^{-z}}{z(\cos{z}-1)}$ at $z=0.$

Let $$f(z)=\frac{e^{-z}}{z(\cos{z}-1)}$$ a) Determine the order of $f$:s pole at $z=0$. b) Find the four first terms $\neq 0$ in $f$:s Laurent expansion valid in the region $\{z:0<|...