Questions tagged [laurent-series]

This tag is for questions about finding a Laurent series of functions and their convergence. 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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26 views

Laurent Series of $\frac{1}{z(1-z)}$ in neighborhood of $z=1$ and $z=0$

So, the question is: Laurent Series of $\frac{1}{z(1-z)}$ in neighborhood of $z=1$ and $z=0$. I know I can find Laurent series' all over MSE, but in an effort to build my own intuition, and to see the ...
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25 views

Determining Laurent series on a region [closed]

$g(z)= \frac { \sinh2z } { (2-z)^2 } $ , $H = \{ z \in \mathbb C : 0 < |z-2| < \infty \} $. Show that $g(z)$ function has Laurent series expansion on $H$ region and determine that Laurent series....
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4answers
76 views

Show that $\sum_{k=0}^{\infty} \frac{k^2+3k+2}{2} z^k = \frac{1}{(1-z)^3}$, without using differentiation

Show that, $$\sum_{k=0}^{\infty} \frac{k^2+3k+2}{2} z^k = \frac{1}{(1-z)^3}$$ where $ z \in \mathbb{C}, |z|< 1$ Well, I have figured out that is a Laurent series I have watched 3 videos in the ...
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18 views

Does a multivariate series expansion exist for this function?

I have the following function that I was hoping to simplify in some way $$\displaystyle f(x,y)=\frac{1+x}{1+y+\sqrt{(1-y)^2+y(1-\frac{1}{x})^2}}$$ defined for $0 < {x,y} \leq 1$ and was wondering ...
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2answers
32 views

Finding the Laurent Series of $f(z)=\frac{1}{(z^2+1)^2}$

I'm trying to determine the residue at $i$ of $f(z)=\frac{1}{(z^2+1)^2}$. My first attempt was to transform this into a Laurent Series: $$f(z)=\frac{1}{(z^2+1)^2}=\frac{1}{(z+i)^2(z-i)^2}$$ ...
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3answers
41 views

Why is $\operatorname{res}(0, \cos(\frac{1}{z}))=0$?

The residue of a function $f$ represented by a Laurent Series in complex analysis is defined as the coefficient at $n=-1$ of the series ($a_{-1}$). Giving the definition of the cossine in complex ...
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1answer
27 views

Solving for x: finite Maclaurin series = finite Laurent series

Given the equation, $$\sum _{i=0}^{n}a_{i}x^{i}=\sum _{j=1}^{m}b_{j}x^{-j}$$ solve for x. I'd typically solve this by multiplying across by $x^{m}$, then shifting everything to one side and finding ...
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3answers
58 views

Is there a series of $e^x$ that only contains $\sin(x)$ in the form of $e^x=\sum\limits_{n=0}^{\infty}c_n\cdot \sin(x)^n$?

Hy i'm trying to find a series of the following kind: $e^x=\sum\limits_{n=0}^{\infty}c_n\cdot \sin(x)^n \ \ \ \ \ \ \ \forall \ x\in \left(a,b \right)$ or maybe $e^x=\sum\limits_{n=0}^{\infty} \sin(...
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1answer
46 views

Residue of $f(z)=\frac{1}{z^2(e^z-1)}$

I want to find the residue of the function $f(z)=\frac{1}{z^2(e^z-1)}$. I have tried something that I am 99 precent sure about, but still I would appreciate some feedback. What I have done is that: $$\...
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1answer
32 views

Laurent Expansion of $\frac{1}{z(z-1)}$ about $0$ using integral form of coefficients

I've seen many questions already on Laurent Expansions and how to find examples like my question, however, most of them turn to geometric series and partial fractions, which I understand can be faster ...
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1answer
29 views

Dirichlet series of complex variable [closed]

I have two Dirichlet's serieses (convergence) that equal on a domain of the complex plane, I want to prove that their coefficients equal, I thought to do it by contradiction but I didn't succeed.
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2answers
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Find a power series that is convergent on the closed unit disk but diverges elsewhere.

Question: Does there exist a power series centered at $z=0$, $f(z)=\sum_{n=0}^\infty a_n z^n$ such that the domain of $f$ is exactly the unit disk $D^2\subset \mathbb{C}$? In other words, I'm looking ...
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1answer
34 views

Calculate the residue of $\exp\left(\frac{z+1}{z-1}\right)$ in every point of $\mathbb{C}$

I have to calculate the residue of $\exp\left(\frac{z+1}{z-1}\right)$ in every point of $\mathbb{C}$. So I tried to compute the Laurent Series expansion $\forall z_0 \in \mathbb{C}$. For $z_0 = 0$ we ...
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1answer
46 views

Finding maximal rings and Laurent Series - solution verification

I am to find regions where $f(z) = \frac{1}{z-2}$ around $i$ has Laurent series. As Jose pointed out, the natural regions are rings $A(i, 0, \sqrt{5})$ and $A(i, \sqrt{5}, \infty)$ because $\sqrt5$ is ...
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43 views

Is there an asymptotic series expansion for $e^{-2\pi ia\log(t)}$ around $t=0$

I'm currently working on a problem which involves the following series: $$f(t)=\sum_{n=-\infty}^\infty a_ne^{-2\pi i\cdot pn\cdot\log(t)}$$ Where $p\in \mathbb R$. I'm interested in the short time ...
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1answer
69 views

What Is the meaning of the residue of a complex function?

Studying complex analysis I saw that, in many cases, the residue's theorem comes really handful. I learned how to find it and how to use it, but I didn't quite understand what it really means "...
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35 views

Laurent series of $f(z) = \frac{1}{z}$ in $A=\{z \in \mathbb{C}: 0 < |z| < R\}$

In complex analysis I learned Laurent series and I was trying to find the Laurent series of some functions as an practice exercise. So I noticed that the Laurent series of the function $f(z)=\frac{1}{...
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27 views

intuition behind the process of finding Laurent series

So I learned in my complex analysis class the concept of Laurent series but my teacher's approach was fully theoretical: no actual example and no example on how to find a Laurent series for a function ...
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1answer
37 views

Let $f$ holomorphic at $D=\{z:0<|z|< 1\}$ Prove that exist a unique $c\in\mathbb{C}$ s.t $f(z)-\frac{c}{z}$ has primitive function.

Let $f$ holomorphic at $D=\{z:0< |z|< 1\}$ Prove that exist a unique $c\in\mathbb{C}$ s.t $f(z)-\frac{c}{z}$ has primitive function in D. So my thoughts were that we can write a laurent series ...
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21 views

$z_0$ is essential singularity if and only $a_k\neq{0}$ for infinitely many negative integers.

Question Let $f:D\left(z_{0},r\right)\setminus\left\{ z_{0}\right\} \rightarrow\mathbb{C}$ be analytic and let $\sum_{k=-\infty}^{\infty}a_{k}\left(z-z_{0}\right)^{k}$ be its laurent expansion. ...
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1answer
53 views

Find the Laurent series for $f(z)=\frac{1}{z(1-z)}$

I am having difficulties finding Laurent series of the above function, around these two domains a) $1<|z|$ b) $1<|z-1|$ For a) I do $$ \sum_{n=0}^\infty \frac{1}{z^{n+1}} - \frac{1}{z} $$ ...
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2answers
89 views

Evaluating: $\lim_{n\to \infty} \sum_{i=1}^n \frac{1}{\frac{an}{b-a}+i}$

$$\lim_{n\to \infty} \sum_{i=1}^n \frac{1}{\frac{an}{b-a}+i},\ a,b\in\Bbb R\setminus\{0\},\ a\ne b$$ I know for a fact that the solution can be found via Laurent Series if that hint helps. I inserted ...
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1answer
56 views

finding a power series for $f(z)=\frac{1}{1+z^2}$ centered at $0$

In an exercise I am asked to find a power series for $f(z)=\frac{1}{1+z^2}$ centered at $0$. My approach was the following: $f(z)=\frac{1}{1+z^2}=\frac{1-z^2}{(1+z^2)(1-z^2)}=(1-z^2)\frac{1}{1-z^4}=(...
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1answer
28 views

Absolute convergence of Laurent series

There is the following result (not proven) on my notes: If $\sum_{j=-\infty}^\infty a_j (z-z_0)^j$ is converges at some point, then there exist $r_1,r_2\geq 0$ (which can be $\infty$) so that the ...
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1answer
52 views

Let $f\in Hol(\mathbb{C}\backslash\{a_1\ldots a_N\})$ where ${a_1\ldots a_N\,\infty}$ are the poles of $f$ show that $f$ is rational function [duplicate]

Let $f\in Hol(\mathbb{C}\backslash\{a_1, \dots, a_N\})$ where ${a_1, \ldots ,a_N,\infty}$ are the poles of $f$. Show that $f$ is rational function. I've tried to define $g(z):=\frac{f(z)}{\prod_{i=1}^...
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27 views

Classification of infinity as a singular point.

i have the followings problems, i must find and classify the singular points of the followings functions: $a) \frac{e^{z}}{1+z^{2}}\\$. $b) \frac{e^{z}}{z(1-e^{-z})}$. $c) \frac{1}{z^{3}(2-\cos{z}...
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1answer
55 views

Use the Laurent series to prove that $\int_{0}^{2 \pi} e^{\cos(t)} \cos(nt-\sin t)\,dt=\frac{2 \pi}{n!}$

Need help to demonstrate the next integral $\int_{0}^{2 \pi} e^{\cos (t)} \cos(nt-\sin t)\,dt=\frac{2 \pi}{n!}$, $n=0, \pm 1,...$ I have tried using Laurent’s development of $e^{1/z}$, but i have ...
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1answer
64 views

What is the residue of $f(z) = z\sin(z + {1 \over z})$ at $z_0 = 0$?

I want to determine the residue of $f(z) = z\sin(z + {1 \over z})$ at $z_0 = 0$. I think that this an essential singularity and I want to work with the Laurent series which should be given by: $$z \...
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1answer
32 views

Determine convergence annulus of Laurent series without computing the Laurent series

Take the holomorphic function: $$\mathbb{C} \setminus \{2k \pi i \ \mid \ k \in \mathbb{Z} \} \ni z \mapsto \frac{1}{e^z - 1} \in \mathbb{C}. $$ How can we determine the annulus of convergence of the ...
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1answer
25 views

Does this strategy of characterizing poles always work?

I stumbled upon a fast way to characterize poles of order $m$ of a meromorphic function $f$ (on some open set $\Omega$) in this answer here. My question is, does this general strategy always work? ...
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1answer
59 views

Finding the Laurent expansion

I have encountered the following question: Question: Consider the Laurent expansion $$ \dfrac{e^{z}}{\cos(2z)} = \sum_{n = - \infty}^{\infty} a_{n}z^{n}, \qquad \dfrac{\pi}{4} < \vert z \vert <...
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25 views

Alternative Laurent expansion of $1-e^{-x}$

I wonder if there is Laurent expansion of $f(x) = 1 - e^{-x}$ where $x>0$, is real number. It is known that the expansion : $$ g(x) = 1- e^{-x} = x - \frac{x^2}{2!} + \frac{x^3}{3!} - ... $$ for $ ...
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0answers
19 views

Showing by Hand that $H^1(0)=0 , H^1(-p)=0$ in $\mathbb{C}$ , Mittag-Leffer Problems in $\mathbb{C}$ and $\mathbb{C}/L$

I am trying to show by hand, without using the Riemann-Roch Theorem, that in $\Bbb C_{\infty}$ we will have that $H^1(0)=0$ and $H^1(-p)=0$, but I am having some trouble constructing the functions ...
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1answer
19 views

Use Laurent Series to Determine Bi-Analytic Function

$w=\varphi(z)$ is bi-analytic. $\varphi: D=\{z \in C: r_1 < |z| < r_2\} \to G=\{w \in C: R_1 < |w| < R_2\}$. $\varphi$ can be written as the following Laurent series $\varphi(z)=\sum_{n=-\...
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1answer
31 views

Laurent series for $ f(z) = \frac{1} {z} + \frac{1} {(1-z)} + \frac{1} {(2-z)} $ around arround $0< |z|<1, 0< |z-1|<1$ and $0< |z|<2$?

Find the Laurent series of the function $$ f(z) = \frac{1} {z} + \frac{1} {(1-z)} + \frac{1} {(2-z)} $$ a)$$ \{z \in\Bbb C: 0<|z|<1\} $$ b) $$ \{z\in\Bbb C:0<|z-1|<1\} $$ c) $$ \{z\in\...
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1answer
31 views

Finding Laurent series of $f(z) = \frac{1}{z^{2}+4}$ on two different domains.

I need to find the Laurent series of $$f(z) = \frac{1}{z^{2}+4}.$$ First for $ z \in \mathbb{C}: |z|<2$ and then for $z \in \mathbb{C}: 1<|z-i|<3$. Now for the first restriction I do the ...
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2answers
30 views

Laurent series of $f(z)=\frac{1}{z^{2}\sinh(z)}$

I would really appreciate if you could help me understand this. So, I’m at this point $\sum_{m=0}^{\infty}\frac{z^{2m+3}}{(2m+1)!}\sum_{k=-2}^{\infty}c_{2k+1}z^{2k+1}=1$ But I don’t really know how ...
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1answer
39 views

Can you find a residue by using Taylor's formula for a single coefficient instead of expanding into a Laurent series?

For example, say I was trying to determine the residue of the functions $\frac{e^z}{\sin z}$ and $-\cot(z)$ at $-\pi$. Clearly there is a singularity in both of these functions at this point, but the ...
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18 views

Find the residue at pole z = 0 of $\frac{1}{z(e^z-1)}$ [duplicate]

I first saw that $z = 0$ is an order 2 pole, then, when I tried to use $\frac{1}{(m-1)!}\frac{\mathrm{d^{m-1}} }{\mathrm{d} z^{m-1}}[(z-z_0)f(z)]$ (evaluated at $z=z_0$) I got such an indetermination, ...
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1answer
23 views

$a_{−3}$ in the Laurent expansion of $f(z)$ in the region $0 < |z| < 1$. $f(z)=\frac{1}{2z^3}-\frac{1}{z^3+i}$

I am looking to find the $a_{−3}$ in the Laurent expansion of $f(z)$ in the region $0 < |z| < 1$. $$f(z)=\frac{1}{2z^3}-\frac{1}{z^3+i}$$ I can't get my head around this Laurent Series: $$\...
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1answer
31 views

Find the Laurent series at z=0

Find the laurent series expansion of $\frac{1}{z(z+5)}$ at z=0. I found the solution for this question in MATLAB. The solution I got was $\frac{5}{z}+1$. I don't understand how the laurent series is ...
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0answers
35 views

Laurent series Approximation in Algebraic Curves

I am reading Rick's Miranda book and he's now talking about how we can do a laurent series approximation in an Algebraic curve,page $173$, that is Suppose that $X$ is an algebraic curve, fix a ...
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2answers
31 views

Finding the limit of a quotient of polynomials containing a constant to the power of $n$

I've been given the infinite series: $$ \sum_{n=0}^{\infty} (n+2^n)z^n $$ I need to find the convergence radius through: $$R = \lim_{n \rightarrow \infty} \left| \frac{a_n}{a_{n+1}} \right|$$ My $...
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2answers
58 views

Series expansion of $\dfrac{z^2}{1+z^2}$

Consider the complex function $$f(z) = \dfrac{z^2}{1+z^2}$$ It has two isolated singularities at $z=\pm i$. So when we consider the series expansion at $z_0 = 0$, then the radius of convergence is $...
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1answer
37 views

Complex integration and Laurent Series

i was asked to calculate the following integral: $$\int_{|z|<1}\frac{1}{z^2\sinh(z)}\: dz$$ I was suggested to calculate the Laurent Series of the integrand function, but how does one determine ...
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2answers
25 views

Calculation of the Laurent Series

How can one calculate the Laurent Series of: $$f(z)=\frac{1}{z(z-i)^2},\quad 0<|z-i|<1$$ So far i have calculated that $$f(z)=\frac{1}{z-i}-\frac{1}{(z-i)^2}-\frac{1}{z}$$
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1answer
26 views

Determine sum function and area of convergence the Laurent series $\sum_{n=1}^\infty \frac{z^{-4n}}{4n} + \sum_{n=0}^\infty \frac{z^{4n}}{(4n)!}$.

I am trying to compute the sum function and area of convergence of $\sum_{n=1}^\infty \frac{z^{-4n}}{4n} + \sum_{n=0}^\infty \frac{z^{4n}}{(4n)!}$, and to determine the singularities of the sum ...
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0answers
25 views

How to find Laurent expansion of $f(z)$

I want to know the Laurent expansion of $f(z)=\frac{z^2}{1-e^z}$ centered at $z=2k\pi i $ with $k\in \mathbb{Z}$ and $k\neq 0$. I tried using the expansion of $e^z$ or the geometric series but with ...
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2answers
24 views

Convergence of Taylor Series (as part of finding the region of Conv for a Laurent series)

I'm given a Laurent series problem and to find the largest region of convergence, I need to fine $R_1$ and $R_2$. The Laurent series is $$z^3 - \frac{z}{3!} + \sum_{k\geqslant2} \frac{(-1)^{k}}{(2k+1)...
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1answer
34 views

Finding the largest region on which Laurent series converges

I need to find the largest region on which $$f(z) = z^4 sin(1/z)$$ defined on $$\mathbb{C}/{0}$$ converges. So for the Laurent series, I got $$z^3 - \frac{z}{3!} + \sum_{k\geqslant2}\frac{-1^{2k+1}}{(...

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