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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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Expansion of the form $\sum_{n=0}^\infty c_n z^n$ [on hold]

write down an expansion of the form $\sum_{n=0}^\infty c_n z^n$ for $\frac{(a+iz)}{a-iz}$
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66 views

What is the Laurent series of $z+(1/z)$? [on hold]

What is the Laurent series of $z+(1/z)$? Is it already in the form of its series?
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0answers
43 views

Principal Part of Laurent Series Converges in Punctured Disc

I'm trying to work through the following problem: Prove that if the holomorphic function $f$ has an isolated singularity at $z_{0}$, then the principal part of the Laurent series of $f$ at $z_{0}$ ...
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1answer
39 views

Finding the laurent series around z = 0

I'm trying to find the Laurent series around $z = 0$ and specify the largest annulus where the expansion is valid. It's for the functions: $f(z) = 1/((z-a)(z-b)) $, for $a,b, \in \mathbb{C}$ and $...
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2answers
140 views

Forming a series representation for $\tan^a(u)$

As part of my attempt at solving this integral, I became stuck with resolve the following definite integral: \begin{equation} \int_0^{\pi/4}u^{b}\tan^a\left(u\right)\,\mathrm{d}u \nonumber \end{...
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1answer
18 views

Parseval identity for Laurent series

Let $0\leq r_1<r_2$, and $z_0\in \mathbb{C}$, and consider the region $A=\{z\in \mathbb{C}|r_1<|z-z_0|<r_2\}$. Let $f$ analytic in the region $A$. Then we can write, $$f(z)=\sum_{n=0}^{\infty}...
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1answer
35 views

On the Newton polygon for Laurent series

I'm stuck with an understanding of what should be the Newton polygon for a Laurent series. I'm reading ''An introduction to G-function" by Dwork and he dedicates only three pages to Newton polygons ...
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3answers
44 views

calcuating residue of a complex function

I need to calculate the residue of the function $\frac{(z^6+1)^2}{(z^5)(z^2-2)(z^2-\frac{1}{2})}$ at $z$=0. z=0 is a pole of order 5 so I tried using the general formula to calculate the residue but ...
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0answers
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Region of convergence in discrete system?

When dealing with a discrete system, it is a widely-accepted idea to use the $z$-transform to analyze properties of the system in frequency domain. However I'm confused of two questions, regarding the ...
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1answer
40 views

Computing the Laurent series of $\frac{1-z^4}{z^4}$ at $z_0=0$

I want to find the Laurent series for $$\frac{1-z^4}{z^4}$$ at $z_0=0$ I would write $$ \frac{1-z^4}{z^4} = \frac{1}{z^4} \frac{1}{\frac{1}{1-z^4}} = \frac{1}{\sum_{k=1}^{\infty} z^{4k}}= a_1z^4 + ...
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1answer
48 views

Do all coefficients in a Laurent series need to be real?

Suppose we are given $ f(x)$ holomorphic in $ \Bbb C \setminus\{0\}$ that has Laurent series around $0$ and $ f(x)\in \Bbb R $ for all $ x \in \Bbb R $, $x\ne0$. Does it imply that all coefficients ...
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1answer
65 views

Laurent series problem on $f(z)=\frac{ z }{ z^2-z-2 }$

I have problems with computing Laurent series of the function $f(z)=\frac{ z }{ z^2-z-2 }\quad$ in the ring centered in $0$ containing point $1+i$. I also have to find the radius of convergence of ...
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2answers
37 views

Complex function to Taylor and Laurent series

I am trying to express a function with Taylor and Laurent series. I've been reading my textbook and also various online resources, but I still can't follow any of the example problems. Here's what I ...
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1answer
17 views

How to prove the convergence region for matrix Laurent expansion?

I know how to do it for scalar Laurent series. However, consider $$\mathbf{F}(z) = \sum_{k=0}^{\infty} C A^k B z^{-k}, $$ where $F, A,B,C$ are matrix with proper dimension. $z \in \mathbb{C}$. I ...
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3answers
42 views

Laurent series in a certain region

The problem: Find the laurent series expansion in powers of $z+1$ of the function: $g(z)= \frac{1}{(z+2)(z-2)}$ For the region $1<|z-1|<3$ My attempt: I suspect I need to do a change of ...
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1answer
30 views

How to handle an identical zero pole when expanding to a Laurent Series?

I have a function given as $f(z)=\frac{2(z-3)}{z^2-8z+15},$ which is clearly the same as $f(z)=\frac{2(z-3)}{(z-3)(z-5)}$ when $z\neq3$. Typically I would break these into two separate fractions using ...
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0answers
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(Inverse) Completion Mapping

Let $\mathbb{F}_q(t)$ be the rational function field and $\mathfrak{p}_\infty$ the place at infinity. I know that every element $z \in \widehat{\mathbb{F}_q(t)}$ (the completion of $\mathbb{F}_q(...
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1answer
33 views

Determining the type of singularities

Determine the type of singularities of $$f(z)=\frac{1}{(z-1)\cot(\pi/z)}\tag{1}$$ We first rewrite the function: $$f(z)=\frac{1}{(z-1)\cot(\pi/z)}=\frac{\sin(\pi/z)}{(z-1)\cos(\pi/z)} \tag{2}$$ ...
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29 views

Infinite products of Laurent series

I am trying to find an expression for the coefficients of a Laurent series which is itself an infinite product of Laurent series: $f(z) = \sum_{u=-\infty}^{\infty} f_{u}z^{u} = \prod_{i=0}^{\infty} \...
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1answer
37 views

Infinite Series arising in Laplace transform

I'm trying to understand how the answer was computed for the following infinite series: $\sum _ { x = 1 } ^ { \infty } e ^ { - s x } p q ^ { x - 1 } = \frac { p e ^ { - s } } { 1 - q e ^ { - s } }$ ...
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1answer
28 views

Calculate the Laurent Series for : $(8z+6+8i)/(2z^2-3z-4iz)$

I have to find the Laurent-Series with $0<|z|<5/2$ and $z_0=0$ for: $$\frac{8z+6+8i}{2z^2-3z-4iz}$$ I already did this: $$ \frac{8z+6+8i}{2z^2-3z-4iz} = \frac{A}{2\cdot(z-0)}+ \frac{B}{2\cdot(...
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1answer
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Finding Laurent series with the use of differentiation

Let $f(z)=\frac{1}{z(z+1)^2}$ Find the Laurent series expansion for f centered at 0 and which converges on ${0 < |z| < 1}$ In my teachers solutions, he has written out the answer as follows: ...
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1answer
34 views

isolated singularity / Laurent series

I want to classify the singularities of $f(z)=\frac{\cos^2 z}{\sin^2 z}$ Maybe I can write: $$f(z)=\frac{\cos^2 z}{\sin^2 z} = \frac{1-\sin^2 z}{\sin^2 z} = \frac{1}{\sin^2 z}-1.$$ I can substitute $\...
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1answer
28 views

Laurent series/isolated singularity

I want to classify the singularities of $$ f(z)=\frac{\sin(2z)}{(z-1)^3}$$ The Taylor series is: $\sin(2z)=\sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)!} 2^{2k+1} z^{2k+1}$ So: $ \frac{\sum_{k=0}^{\...
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1answer
41 views

Is the textbook solution wrong by a sign? Laurent Series

Find the Laurent series of $\frac{e^z}{z^2 -1}$ about $z = 1$. Here is my solution: Factor denominator $\frac{e^z}{(z-1)(z+1)}$ let $w = z - 1$, and so $z = w + 1$, substitute in $\frac{e^{w+1}}{w(w+...
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0answers
31 views

Laurent series $\frac{e^z}{z^2 -1}$ about $z=1$

Where to start? I know I need to make all the z's into $ z-1 $'s, so $ e\cdot e^{z-1} $ and denominator into $ (z-1)(z+1) $, but from there what do I do with the denominator?
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1answer
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Find order of pole $\frac{e^z -1}{z^2 +4}$, about $z=2i$.

$$\frac{e^z -1}{z^2 +4},\quad\text{about $z=2i$.}$$ The textbook I'm reading isn't specific about these case, only gives basic examples. Basically to find pole I'd have to expand a Laurent series ...
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1answer
30 views

Expanding Laurent Series with $(z-1)$ term on numerator

$\frac{z-1}{z^3\cdot(z-2)}$. I want to expand series in regions $|z|<2$. I tried to just pull out $\frac{(z-1)}{z^3}$ and expand the $\frac{1}{(z-2)}$,the answer I got is similar to solution but ...
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1answer
18 views

Classification of isolated singularity /Laurent series

I want to find out, what kind of singularties does $ f(z)= \frac{1}{z^3-z^5}$ have. I would do the following steps: $ f(z)= \frac{1}{z^3-z^5} = \frac{1}{z^3(1-z)(1+z)}$ so I have $ z_1=1, z_2=-1 , ...
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2answers
33 views

Expand complex function into series (factoring not working)

$\frac{z}{z^2 + 9}$, I need to turn into series, where z = complex number. I have tried factoring denominator to get $\frac{1}{z} \cdot \frac{1}{(1 + \frac{9}{z^2})}$ and use binomial expansion. But ...
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1answer
22 views

Expand a complex function into a series

I need to expand this function $\frac{1}{2i+z}$ into a series, z is a complex number. I initially tried substituting $z = x+iy$ into z, combine imaginary terms, then use binomial theorem. However, the ...
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3answers
70 views

Why does $\sin\left(\frac{\pi}{z^2}\right)$ not have a Laurent series at $z=0$?

I tried to compute the Laurent series of $\sin\left(\frac{\pi}{z^2}\right)$ at $z=0$, then I realized something was off altogether and there simply doesn't exist a Laurent espansion centered at $z=0$ ...
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1answer
35 views

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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1answer
33 views

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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1answer
38 views

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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1answer
52 views

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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1answer
36 views

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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1answer
37 views

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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1answer
47 views

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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1answer
88 views

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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1answer
42 views

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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1answer
38 views

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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1answer
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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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2answers
40 views

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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1answer
59 views

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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3answers
59 views

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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0answers
24 views

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 $...
2
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1answer
53 views

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 ...
4
votes
3answers
63 views

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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1answer
47 views

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 ...