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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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 = ...
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27 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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25 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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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 ...
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1answer
34 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 ...
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30 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 ...
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47 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
41 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
31 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 ...
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44 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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19 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
51 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
17 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
25 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 ...
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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
19 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 ...
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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 ...
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1answer
30 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 ...
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1answer
27 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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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 ...
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1answer
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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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34 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
59 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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24 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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77 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: ...
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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
26 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 ...
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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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36 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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33 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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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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34 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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$g(z)=\frac{1}{3-z}$ - 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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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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15 views

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
33 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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2answers
36 views

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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1answer
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 ...
3
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1answer
27 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
30 views

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
36 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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1answer
39 views

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
30 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 ...