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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Which function of $x$, other than $x +c$, and Integral of ($\cos x)^2+(\sin x)^2$ have derivative =$1$

It is a simple question:Which function of x, other than x +c, and Integral of (cosx)^2+/(sinx)^2 have derivative =1. Alternative question, Which other equation gives its derivative as a real number. ...
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0answers
13 views

Determine the nature of the isolated singularities of $f$

I'm learning complex analysis, specifically Laurent series and isolated singularities, and need to verify my work to this exercise since my textbook does not provide any solution: Find and ...
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1answer
23 views

Finding the Laurent series of a function

I'm trying to work through the following example: Find the Laurent series of: $$ f(z) = \frac{1}{z(z-2)^3}, $$ about the singularities $z = 0$ and $z = 2$ (separately). Hence ...
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3answers
44 views

How to determine Laurent series associated to $f(z)$ [on hold]

The function is $$f(z)= \frac{1}{(e^z -1)},$$ $z$ belong to $\mathbb{C}$ and $0<|z|<1$. I need a general expression in term of a sum from 0 to infinity
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1answer
36 views

Calculate the residues of this complex function

Calculate the residues of this complex function $$\frac{1}{z^2\sin(z)}$$ I can notice that we have singularities at $z=n\pi$, where $n=0,1,2,3,\dots$ But, how to find the residues?
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28 views

Determine the nature and situation of the singularities of this function [closed]

Determine the nature and situation of the singularities of this function $f(z) = \frac{1}{z(e^z -1)}$ and show their residues.
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0answers
29 views

Laurent-Series on an annulus

I solved an exercise and I want to know if it is correct. I'm trying to find the Laurent-Series for $$g(w)=\frac{w}{1+w^2}$$ On the annulus $D_{1,2}(-i)$ What I did so far: We have 2 poles in $w=i$ ...
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0answers
42 views

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

The exercise: Find the laurent expansion of $f(z)=\frac{z}{z^2+1}$ in $K_{1,2}(-i)$. My thoughts: $K_{1,2}(-i)$ denotes the annulus. 1 and 2 the radiuses. First thing I did is decompose in partial ...
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1answer
34 views

Radius of convergence and the existence of antiderivative

I think I have some misunderstandings regarding some basic concepts. First, the question I'm dealing with is the following: Let $f$ be analytic in $\{z ;|z|>1 \}$, and $\int_{|z|=2}f(z)dz=0$. ...
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1answer
42 views

Find and classify isolated singularities of $f(z) = \frac{z}{1-e^{z^2}}$ and calculate residues on them

Obviously, all isolated singularities will be of the form $z=\sqrt{2 \pi i k}$ for $k \in \mathbb{Z}$ but I don't know how to classify. I tried expanding $\frac{1}{f} = \frac{1-e^{z^2}}{z}$ to $- \...
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2answers
133 views

Analytic continuation on the disk

Let $V = \big\{z: |z|<5,\text{Im}(z)>0 \big\}$. Let $f$ analytic in $V$, continuous in $\overline{V}$ and suppose $$\forall x \in \left[ -5,5\right]:\ f\left( x\right) \in \mathbb{R}$$ Show that ...
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1answer
46 views

Singularities in the complex plane and expansion of Taylor/Laurent Series

The function f(z) = $\frac{\cosh(z-3i) -1}{(z-3i)^{5}}$ has one singular point in $\mathbb{C}$. I understand that the singular point is an isolated singularity at 3i, and I know there are certain ...
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0answers
16 views

Series representation for Bessel functions

Let $\lambda\in\mathbb{C}$. I want to know if the following is correct: $$\exp\left(\frac{\lambda}{2}(z+z^{-1})\right)=\exp\left(\frac{\lambda}{2}z\right)\exp\left(\frac{\lambda}{2}z^{-1}\right)=\sum_{...
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0answers
20 views

Series representation of $1/\pi\int_0^\pi e^{\lambda \cos(t)}\cos(nt)dt$

Are there representations of $a_n=1/\pi\int_0^\pi e^{\lambda \cos(t)}\cos(nt)dt,~n\in\mathbb{N},~\lambda\in\mathbb{C}$ as a series? These $a_n$ are the coefficients of the Laurent series $a_0+\sum_{n=...
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1answer
34 views

Laurent series of $\exp({\frac{\lambda}{2}(z+z^{-1}))}$

Let $\lambda\in\mathbb{C}$. I want to get the laurent series of $f(z)=\exp({\frac{\lambda}{2}(z+z^{-1}))}$ on $C(0;0,\infty):=\mathbb{C}\setminus\{0\}$ as $$a_0+\sum_{n=1}^{\infty}a_n(z^n+z^{-n})$$ ...
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1answer
35 views

Find a Laurent expansion

I need to find Laurent expansion for function $$f(z) = \frac{iz^2 + 4iz + 4 +12 i}{(z^2+4)(z+2-i)}$$ for $2 < |z-i| < 3$. I start with division: $$\frac{iz^2 + 4iz + 4 +12 i}{(z^2+4)(z+2-i)} = \...
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1answer
23 views

Find the Laurent Series for $\frac{z-2}{z+1}$ around $z = -1$

Find the Laurent Series for $\frac{z-2}{z+1}$ around $z = -1$ I'm not sure how to do this because it is not something with a simple numerator. If it was something like $\frac{1}{(z-2)(z+1)}$ I ...
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2answers
79 views

Proving that two branch cuts can cancel out

Define the following functions $\mathbb{C}\to\mathbb{C}:$ $$u(z)=\frac{\log \left(z+\frac{1}{2}\right)}{z}\quad \left[-\pi\leqslant\arg \left(z+\tfrac12\right)<\pi\right];\quad v(z)=\frac{\log z}{z}...
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36 views

Laurent Series, How it is done

Suppose that a series $$\sum_{n=-\infty}^{\infty}x[n]z^{-n}$$ converges to analytic function $X(z)$ in some annulus $R_1<|z|<R_2$. That sum $X(z)$ is called the z-transform of $x[n]$ $(n=0,\mp 1,...
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1answer
72 views

Laurent series of $\frac{e^z}{z^2+1}$

I cant figure out the laurent series of the following function. Let $f(z)= \frac{e^z}{z^2+1} $ and $|z|\gt 1$ $$\frac{1}{z^2+1}=\sum_{n=0}^{\infty}(-1)^nz^{-2n-2}$$ and $$e^z = \sum_{n=0}^{\infty}...
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36 views

Radius of convergence of the series $\sum\limits_{-\infty}^{\infty}(2^{-n}+4^{-n}) z^n$

I'm trying to find for what values of $z\in\mathbb{C}$ the series $$\sum_{n=-\infty}^{\infty}(2^{-n}+4^{-n})z^n$$ converges. My main methods are the nth root test and the ratio test. I believe it can ...
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33 views

When finding Laurent series when to use partial fractions?

When finding the Laurent series of $$f(z):=\frac{1}{z(z-1)(z-2)}$$ valid in the region $1<|z-2|<2$ for example do we just use partial fractions to break $f(z)$ up and the just find the Laurent ...
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1answer
49 views

How to find the Laurent series of $\frac{1}{z^4(1-z)^2}$ for |z|>1?

A hint is given that $$\frac{1}{(1-\frac{1}{z})^2} = \frac{z^2}{(1-z)^2}$$ and we know that $$\frac{1}{1-w} = \sum_{n=0}^{\infty} w^n$$ for $|w|<1$. I don't know how to make $\frac{1}{z^4(1-z)^...
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1answer
38 views

Verification on finding the radius of convergence of a Laurent series, “the largest R”.

Question: Determine the largest number $R$ such that the Laurent series of $$f(z)= \dfrac{2}{z^2-1} + \dfrac{3}{2z-i}$$ about $z=1$ converges for $0<|z-1|<R$? Attempt: The radius of ...
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33 views

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

I am trying to find the Laurent series of the function $$f(z)=\frac{1}{z(z-1)(z-2)}$$in the rings: 1) $0<|z-1|<1$, 2) $1<|z-1|$, 3) $1<|z-2|<2 $ First I expressed $f$ as $$f(z)=\...
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1answer
50 views

Determineing the largest number such that the Laurent series of converges for a trig function.

Question How to determine the largest number $R$ such that the Laurent series of $$f(z)= \dfrac{2sin(z)}{z^2-4} + \dfrac{cos(z)}{z-3i}$$ about $z=-2$ converges for $0<|z+2|<R$? Attempt : Its ...
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1answer
43 views

How to find the Laurent series expansion of an exp function.

Question: How to find the Laurent series expansion in powers of z of a) $f(z)= \dfrac{e^{z^2}}{z^3}$ $\text{where} \left| z \right| > 0$ Attempt: I know that the main idea is to ...
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1answer
37 views

Laurent series expansion of $\frac{z^2-1}{z^2+1}$

Given $f(z) =\frac{z^2-1}{z^2+1}$, I need to find it's Laurent series expansion at open disk $\sqrt{2} < |z-1| < +\infty$ So at first I've found that, at $z=\pm i$ function is not defined....
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54 views

Construction of function holomorphic in $\mathbb{C}\setminus\{0, 1\}$ satisfying specific conditions

I'm learning about complex analysis, specifically (Laurent) series and residues, and need help with the following problem: Construct a function $f(z)$ holomorphic in $\mathbb{C}\setminus\{0, 1\}$ ...
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23 views

Laurent series of logarithm

Lets have a function $$f(z)=\ln(\frac{z-a}{z-b})$$ on the region where it is holomorphic(off course). I want to find the laurent series for this function. Now finding the taylor expansion of this ...
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2answers
55 views

Laurent Series of $1/\tan z$ [duplicate]

How can we find the Laurent series of the function $$f(z)=\frac{1}{\tan z }$$ around 0. Thank you very much.
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1answer
24 views

Finding the Laurent series expansion of $f(z)$

Find the Laurent series expansion of $f(z)=\frac{z^2-1}{(z+2)(z+3)^2}$ at $0<|z+3|<1$ I have a couple of doubts in how to handle this problem: First of all, should I do it with partial ...
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1answer
26 views

Laurent series expansion of $f(z)=\frac{1}{z(z-1)(z-2)}$

Find the Laurent series expansion of $f(z)=\frac{1}{z(z-1)(z-2)}$ for $1<|z|<2$ So I started of by doing partial fractions: $$f(z)=\frac{1}{2z}-\frac{1}{z-1}+\frac{1}{2(z-2)}$$ First of all ...
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1answer
39 views

Show that if $ |f( \frac{1}{n}) | \leq \frac{1}{n!}$ then $0$ is an essential singularity

Given holomorphic non-constant function $f:D(0,1) \smallsetminus \{0\} \rightarrow \mathbb{C}$ so $\forall n=2,3,...:\ |f(\frac{1}{n})| \leq \frac{1}{n!}$ I need do show that $0$ is an essential ...
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1answer
20 views

Laurent series about singular point for: $\left( \frac{1+x}{1-ax}\right)^{\frac{1}{3}}$

$\left( \frac{1+x}{1-ax}\right)^{\frac{1}{3}}$ I wish to find the Laurent series about the singular point $x=1/a$. I can find an expansion for the left side ($x=0$) and the right side ($x \rightarrow ...
2
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1answer
39 views

Finding a general coefficient within a Laurent expansion

Find a general expression $c_n$ in the Laurent series $$f(z)=\sum_{n=0}^\infty c_nz^n$$ for the function $f(z) = e^{z+1/z}$. So I have \begin{align} f(z) &= \sum_{i=0}^\infty \frac{z^i}{i!}...
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1answer
36 views

Residue of a function sin(1/z)/(1-z)

How do I find the residue of a function $\sin(1/z)/(1-z)$ at $z=0$ ? $z=1$ is a simple pole and residue of this function is $-sin1$. When at I write Laurent series of $\sin(1/z),$ $z=0$ appears to ...
2
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1answer
29 views

Calculating Laurent series for log function

Suppose I have the function: $ f(z) = \frac{log(z+1)}{z^3} $ I need to find the Laurent series of this function to determine its order. In this case we take $log(z)$ to be $ln|z|+i\arg (z)$ So we ...
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2answers
18 views

Finding the first four terms of the Laurent series

I want to find the first 4 terms of the Laurent series expansion of the expression $\frac{1}{e^z-1}$, around $z_0=0$. I tried expanding first $e^z-1=(1+\frac{z}{1}+\frac{z^2}{2!}+\frac{z^3}{3!}+\frac{...
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1answer
15 views

Residue of $\dfrac{1}{z^2+4z+1}$. Laurent series.

I want to calculate : $$ \int_0^{2\pi} \dfrac{\mathrm{d} \theta}{2+\cos(\theta)} $$ I use $z=\mathrm{e}^{\mathrm{i} \theta}$ and residue theorem : $$\int_0^{2\pi} \dfrac{\mathrm{d} \theta}{2+\cos(\...
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1answer
40 views

Residue of $1/(\sin(1/z))$ defined at $z=0$? Trying to derive Laurent Series of $\csc (1/z)$ to find it.

This question is related to this one. I was able to figure out on my own that the residue of $\displaystyle \sin \left(\frac{1}{z} \right)$ is defined at $z=0$ by finding the Laurent Series of $\...
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1answer
43 views

Classifying singularities and determing orders of complex functions

Here are a few functions for reference purposes: $f(z) = \frac{sin(2z)}{z^3}$, $ \space g(z) = \frac{sin(z)}{tan(z)}$, $ \space h(z) = z^2 sin(\frac{1}{z})$ Suppose I was calculating the ...
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0answers
29 views

What is the explicit expression for the Laurent series?

Is the following an explicit expression for the Laurent series: $$f(z)=\sum_{n=0}^{\infty}(z-z_0)^n\frac{f^{(n)}(z_0)}{n!}?$$ The reason I ask is because this is what I have seen being derived when ...
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16 views

Method for determining where Laurent series converge

I have to find the Laurent series for $f(z) = \frac{1}{z^2(4z-1)}$. I know there are two series, centered at 0 and at 1/4, because that is where $f$ is not analytic, and I found the series using the ...
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1answer
62 views

Laurent series for $f(z) = \exp(z+\frac{1}{z})$ around $0$

I need to find the Laurent series of the following function around $0$ - $$f(z) = \exp(z+\frac{1}{z})$$ Now by power series expansion, I got $$f(z) = \sum_{m=0}^{\infty} \frac{z^m}{m!} \sum_{k=0}^{\...
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2answers
25 views

Finding Possible Meromorphic functions on $\mathbb{C}$

I am trying to find all meromorphic functions on $\mathbb{C}$ such that: $$ \mid f(z) \mid \leq (\frac{3 \mid z \mid}{\mid z + 1 \mid})^{3/2}$$ Can I express the functions as: $$f(z) = \frac{C_{-2}}...
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2answers
78 views

Finding the Laurent Series for $\frac{1}{e^z-1}$ for $0<|z|<2\pi$

Since $\left|\dfrac{1}{e^z}\right|<1$ I figured I could rewrite the given function into a geometric series: $$\sum_{n=1}^{\infty} \frac{1}{(e^z)^n}$$ But this seems to be way off the mark. I ...
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1answer
28 views

Find the first terms of the Laurent series for: $\frac{e^{\frac{1}{z}}}{z^2-1}$

$\frac{e^{\frac{1}{z}}}{z^2-1}$ for $|z|>1$ I factored out the denominator and rewrote it to a geometric series and got the following expression: $$e^{1/z}\sum_{n=0}^{\infty}\frac{1}{(z^2)^{n+1}}...
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22 views

Difficulty in Laurent series

I have to find the order of pole of $$f(z)=\frac{\sinh z}{z^7}$$ after expansion of this function I get $$f(z)=1/z^6+1/3!z^4+1/5!z^2+1/7!+z^2/9!$$ It contains only three term in the ...
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1answer
32 views

What is the Laurent expansion of $f(z)=\frac1{z-3}$?

What is the Laurent expansion of $f(z)=\dfrac1{z-3}$? In the region, $|z-3|>0$ ? I just computed the Laurent expansion in the region $|z|>3$ by dividing the denominator by $\dfrac1z$ and ...