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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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} ...
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27 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
40 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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28 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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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
54 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!} ...
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13 views

Uniformly convergent Laurent Series [closed]

Why does a Laurent Series with positive and negative parts converge uniformly only on compact sets?
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22 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) = ...
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2answers
72 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
26 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: ...
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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
15 views

What is the Laurent expansion of f(z)=1/(z-3)?

What is the Laurent expansion of f(z)=1/(z-3)? In the region, ㅣZ-3ㅣ>0 ? I just computed the Laurent expansion in the region ㅣZㅣ>3 by dividing the denominator by 1/z and making it as a geometric ...
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1answer
24 views

Expand the Laurent series

Expand $f(z)= \frac {z}{(z+1)(z-2)}$ in a Laurent series valid for the given annular domain: $0 \lt \lvert z+1 \rvert \lt 3$ I'm having a lot of trouble with this one. The answer, per the back ...
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35 views

Using the $\cot (\pi z)$ to find $\sum \frac{1}{n^2}$ [duplicate]

I'm trying to prove the result that $$\sum_1^\infty \frac{1}{n^2}=\pi^2/6$$ using cotangents and residue theory. I know that $\sum f(n)=-$Sum of residues of $\pi \cot (\pi z)f(z)$ at the poles of ...
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30 views

Determine the Laurent expansion of $f(z)=\frac{z}{(z-1)(2-z)}$ for different regions in the complex plane.

I have tot determine the Laurent series of $f(z)=\frac{z}{(z-1)(2-z)}$ for the regions $|z-1|>1$ and $0<|z-2|<1$. I already know what to do for the regions $|z|<1$, $1<|z|<2$ and ...
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22 views

principle laurent series of $f(z)=\frac{(e^z-1)(1-\cos(2z))}{z^4\sin(z)}$ at $z=0$ and determine $\oint_{|z|=1} f(z)dz$.

Question: So given the function $$f(z)=\frac{(e^z-1)(1-\cos(2z))}{z^4\sin(z)}.$$ First: Give the principal part of the Laurent series of $f$ at $z = 0$. Second: Determine the integral $\oint_{|z|=1} ...
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1answer
28 views

Finding Laurent series

I'm having trouble in finding the Laurent Series of this function: $f(z)=\frac{1-z}{(1-2z)^2}$ Near the point $z=\frac{1}{2}$ I know the answer from Wolfram Alpha, but I don't understand how to get ...
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22 views

$g(z)=\frac{1}{3-z}$ - Laurent series for two differents annulus

Find the Laurent series for the function $g(z)=\frac{1}{3-z}$ for the annulus $0 < |z|<3$ and $|z|>3$. I understand for the first case, the Laurent series would be $\sum_{n \geq0} ...
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27 views

Find and classify singular points of $\cot\left(\frac{1}{z}\right)$

I need to find and classify singular points (i.e., decide whether the point is removable, a pole of order $N$, essential, or not an isolated singular point), including infinity, of ...
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$f$ has pole of order $m$ and $g$ has a pole of order $n$ at $z_{0}$, show $f+g$ has isolated singular point there

I am faced with the following problem: Suppose $f(z)$ and $g(z)$ have poles of order $m$ and $n$ respectively, at a point $z_{0} \in \mathbb{C}$ with $m \neq n.$ Show that $z_{0}$ is an isolated ...
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Branch dependence on Laurent Series?

Consider the function: $$f(z)=\ln(1-z)$$ in the principal branch (i.e. with $arg(1-z)\in [-\pi, \pi]$) the Laurent series of this function is: $$\ln(1-z)=-z-\frac{1}{2}z^2-\frac{1}{3}z^3...$$ My ...
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Determine the singular part of $f$ at these poles.

Show that $f(z)=\tan(z)$ is analytic in $\mathbb{C}$ excpet for simple poles at $z=(n+\frac{1}{2})\pi$ for $n \in \mathbb{Z}$. Determine the singular part of $f$ at these poles. My thought is to ...
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2answers
20 views

How to find the largest $R$ such that the Laurent series of $f(z)=\frac{2}{(z^-1)}+\frac{3}{2z-i}$ about $z=1$ converges for $0<|z-1|<R$?

How to find the largest $R$ such that the Laurent series of $$f(z)=\frac{2}{(z^2-1)}+\frac{3}{2z-i}$$ about $z=1$ converges for $0<|z-1|<R$? What I have done so far: ...
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3answers
31 views

Iteration of $A_{n}(q)=q^nA_{n-1} (q)$

I can't seem to find how $A_{n}(q)=q^nA_{n-1} (q)$ iterates to $$A_{n}(q) = q^{n+1 \choose 2}A_{0}(q)$$ Where ${a \choose 2} = a(a-1)/2$ and absolute value of q is less than 1. I understand that I ...
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Laurent Series Expansion Logic

Relative to the below image, I am curious about the progression from equation 3.2 to equation 3.3, then from equation 3.3 to equation 3.4. I understand the logic in 3.2. I understand that a Laurent ...
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37 views

Laurent series for $z^{2} e^{1/z}$ at $z = \infty$

I just found the Laurent series for $z^{2}e^{1/z}$ for $z = 0$, and now I need to find it at $z = \infty$. (for $z=0$, it was $\displaystyle \sum_{n=0}^{\infty}\frac{z^{2-n}}{n!}$, by the way). I'm ...
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1answer
26 views

Laurent expansion of $\frac{1}{(z-a)^{k}}$, $k \in \mathbb{N}$

I need to expand the function $f(z)=\frac{1}{(z-a)^{k}}$ where $a \in \mathbb{C}$, $a \neq 0$, $k \in \mathbb{Z}$, $k>0$ in a Laurent series in the annuli (a) $0< |z|<|a|$ (b) $|a|<|z|$ ...
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2answers
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Expand the function $f(z)=\frac{1}{(z-a)(z-b)}$ where $0 < |a| < |b|$ in a Laurent series in different annuli

I have to expand the function $f(z) = \frac{1}{(z-a)(z-b)}$ where $a, b \in \mathbb{C}$, $0 < |a| < |b|$ in the following annuli: (a) $0<|z|<|a|$ (b) $|a|<|z|<|b|$ (c) ...
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1answer
34 views

Laurent Series of $(z-2)/(z+1)$ at $z=-1$ [closed]

What's the Laurent series expansion of $\frac{z-2}{z+1}$ at $z=-1$?
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What is the purpose of this manipulation?

In my complex analysis class, we often perform a transformation to investigate the singularities of a function. For example, given $f(z)=\cot(1/z)=\frac{\cos(1/z)}{\sin(1/z)}$ where we find that ...
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1answer
29 views

Determine the Laurent Series

Determine the Laurent Series Expansion for the function $$(z^2-1)\exp\left(\frac{2z}{z-1}\right)$$ at $a=1$ and find its ring of convergence. I expanded the exponential function but I couldn't get ...
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1answer
25 views

Laurent series of $f(z) = \frac{1}{z^2-z}$ centered at $z= -1$ and converges at $z=-1/2$

I need to compute the Laurent series expansion of the following function - $$f(z) = \frac{1}{z^2-z}$$ centered at $z= -1$ and converges at $z=-1/2$ . I tried this problem using substitution $w=z+1$. ...
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2answers
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Uncertain how the following step was accomplished.

I'm working through a book example that aims to find the first two nonzero terms of the Laurent expansion of $f(z)=\tan(z)$, about $z=\frac{\pi}{2}$. The substitution $z=\frac{\pi}{2}+u$ is made ...
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1answer
46 views

Calculating residue of $f(z)$ around $a=\infty$

Let $f(z) := \frac{3\sin 4z}{z^4}$. Find $$\mbox{res}\left [f(z),\infty\right ]$$ For $a=\infty$, the residue is given as $-\frac{1}{2i\pi}\int_{|z|=R} f(z)\mbox{d}z$. $$\int_{|z|=R} \frac{3\sin ...
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1answer
49 views

the laurent series expansion [closed]

Please, I need someone to help me to solve this problem, thanks. Let $$f(z) = \frac{2z+1}{z(z+i)},$$ Find the laurent series expansion for $f(z)$ in $ann(0;1,∞).$ Find the laurent series expansion ...
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Classification of singularities in function

Classify the singularities (removable, poles, essential) of the function $$f(z) = \dfrac{\sin\left(\dfrac{z}{z+1}\right)}{z(z-1)^2(z^2+1)}$$ First we consider the isolated singularity at $z=0$. ...
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1answer
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Determine the nature of an isolated singularity

Using the Laurent series, determine the nature of the isolated singularity $z = \pi$ for the function $$f(z) = \frac{1-e^{-z}}{\sin{z}}$$ $$e^{-z} = \sum_{n=0}^\infty ...
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Laurent Expansion Theory

If $f$ has a Laurent expansion like this: $$ f(z) = \sum_{n = -\infty} ^{\infty} a_n(z-z_0)^n$$ in the annulus $r < |z-z_0|<\infty$, then, I need to prove that the function $f$ can also be ...
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1answer
18 views

Coefficients of a certain Laurent series

If $$e^{t(z - 1/z)/2} = \sum_{n \in \mathbb{Z}} J_n(t)z^n$$ is the Laurent expansion in $\mathbb{C}^*$, how can I show that $$ J_{-n}(t) = (-1)^n J_n(t)$$ Any help would be appreciated.
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1answer
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Classify the singularity - $f(z)=\dfrac{z}{z^2+\sin^3z}$

Classify the singularity $z_0=0$ of $f(z)=\dfrac{z}{z^2+\sin^3z}$. I found that $$ \lim_{z \to 0} zf(z) =\lim_{z \to 0} \dfrac{z^2}{z^2+\sin^3z}=1, $$ so I have a pole of order $1$. Am I right? ...
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25 views

Expanding $1/z$ about $z=-1$ using Taylor series vs Power Series

I need to expand $1/z$ about $z_0=-1$. I decided to do it using both methods, which don't agree. Using Taylor: Finding coefficients: $$f^{(n)}(z)=(-1)^n n!/z^{n+1} \Rightarrow f^{(n)}(-1)=-n!$$ ...
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Invertible matrices over ring of formal Laurent series

Let $A$ be a commutative ring, let $A[[t]]$ be the ring of formal power series and consider the ring of formal Laurent series $A((t)) = A[[t]][t^{-1}]$. I would like to know: What is ...
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1answer
64 views

Calculating a certain Laurent series (two parts)

I'm trying to solve the following problem: Let $t \in \mathbb{R}$ be fixed and let $$ e^{[\frac{t(z - 1/z)}{2}]} = \sum_{n \in \mathbb{Z}} J_n(t)z^n$$ be the Laurent expansion in $\mathbb{C}$*. ...
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1answer
31 views

Laurent series expansion of function $f(z) =\frac{z^2+1}{z(z-3)}$

I am given the problem to calculate the Laurent series expansion fo $f(z) = \frac{z^2+1}{z(z-3)}$ in the regions $0<|z|<3$ and $3<|z|< \infty$. My question is around what center should I ...
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1answer
23 views

Determine principal part of Laurent' series around point $a$

Let $$f(z) = \frac{z+1}{\sin ^2z}, a=0;$$ First we find the series for $\sin ^2z$: $$\sin^2 z = 1-\cos^2 z =\frac{1}{2}(1-\cos 2z)= \frac{1}{2}\left (1-\sum_{n=0}^\infty ...
2
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3answers
34 views

Laurent series of function, partial fraction decomp. problem

Find the Laurent series of $f(z)$ in the region $1<|z-i|<\sqrt{2}$ $$f(z) = \frac{1}{z(z-i)^2(z-1)} $$ The region doesn't include any of the singularities hence expansion is possible The problem ...
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1answer
23 views

Annular regions for which this Laurent series converges

Given the Laurent series $$\sum_{n= - \infty}^{\infty} \frac{z^n}{3^n + 1}$$ Find the annular region for which it converges. I'm struggling to find any similar examples or where to begin for this.
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1answer
26 views

Taylor vs Laurent series - cosines and sines

In general, why do we say that the Taylor series of sines and cosines are also Laurent series despite of the power of $z$?
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33 views

$f(z)=\frac{\sin z}{z^2}$ - Two different Laurent series

I have to find the Laurent series of the function $f(z)=\frac{\sin z}{z^2}$ in these open annulus $0 < |z|< 3$ and $|z|>3$. Is anyone able to give me the difference between these two series? ...
4
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1answer
21 views

Order of a zero - $\frac{1}{e^z-1}$ - Equivalence between two definitions

I had to classify the singularity (removable, pole and essential) of $\displaystyle \frac{1}{e^z-1}$. I know that $e^z-1=0 \iff e^z=1 \iff z = 2\pi k i = z_k$ for each $k \in \mathbb{Z}$. In using ...