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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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An analytic function $f(z)$ on $U$ can be decomposed as $ f(z)=f_1(z)+f_2(z) $

Let $C_1$ and $C_2$ be simple closed curves in $\mathbb C$ and assume that $C_2$ is in the interior of $C_1$. Let $U$ be the region bounded by $C_1$ and $C_2$. Prove that an analytic function $f(...
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Why can't we multiply two Laurent series

Wikipedia claims that Laurent series cannot in general be multiplied. I am wondering why not? Suppose we have $f(z),g(z)$ analytic in the annulus: $r<|z-a|<R (0\le r<R\le\infty)$, then $$ f(z)...
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Rewriting a state as a field in CFT

I've been working through a textbook and course on conformal field theory recently. However in a section illustrating how to calculate correlators for secondary fields (using the free boson as an ...
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Residue form an observation

Let $V(z)$ vector space spanned by $e(k),o(k)$. We define them as follows. $$e^k(z):=\Bigg(\frac{\partial}{\partial z}\Big(z\frac{\partial}{\partial z}\Big)^{2k}\frac{z^2}{(1-z^2)}+\delta_{k,0}\...
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What is the Laurent expansion of $\frac{e^{1/z^2}}{z-1}$?

As above, I do not know how to get the Laurent expansion of $\frac{e^{1/z^2}}{z-1}$ over 0. What I think I understand is splitting the denominator into $$ \frac{1}{z}\sum_0^\infty \frac{1}{z}^n + \...
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What happens to $z\in\overline{D_R(z_0)}$?

In my german textbook on complex analysis (K. Fritzsche: Grundkurs Funktionentheorie) in the context of Laurent Series there is the following theorem stated (translated by me, so all errors are due to ...
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Laurent Series of a Rational Polynomial

I am trying to compute the Laurent expansion of $\frac{x}{x-1}$ around $\alpha=2$. I wrote it out as a partial fraction expansion so we have: $$ \frac{x}{x-1} = 1 + \frac{1}{x-1} = 1-\frac{1}{1-x} $$ ...
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Laurent series of a cosine function

I want to find Laurent series of the complex function $$f(z) = \cos\left(\frac{z^2-4z}{(z-2)^2}\right)$$ at $z_0=2$. I will be thankful for any hints.
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Computing the $\operatorname{res}_{z=0}\frac{z^{n-1}}{\sin^n z}$ via an aproximation

Compute the $$\operatorname{res}_{z=0}\frac{z^{n-1}}{\sin^n z}\:\:\text{for}\:\:n\in\mathbb{N}$$ I was given the hint that $\frac{z^{n-1}}{\sin^n z}\sim\frac{1}{z}$ as $z\to 0$. However I am not ...
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Finding the Laurent series for a function with arbitrary values.

The problem is: Let a and b be complex numbers such that $0<|a|<|b|$. Find a series in positive and negative powers of z that represent the function $\frac{1}{(z-a)(z-b)}$ in the annulus $|a|<...
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Is my laurent series expansion for $f(z)=\frac{e^z}{(z-1)^2}$ for $|z-1|>0$ about $z=1$ correct?

Hi could anyone help me with this past exam question (we weren't provided answers) Find the Laurent series expansion of $f(z)=\frac{e^z}{(z-1)^2}$ for $|z-1|>0$ about $z=1$ I have tried to ...
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Errant Minus Sign in a Residue Computation

I'm working on the following computations in a book: Use Cauchy's residue theorem to evaluate the integral of each function around $|z| = 3$ in the positive sense. $$\frac{e^{-z}}{(z-1)^2}$$ I ...
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Computing $\int_{|z-i|=\frac{3}{2}}\frac{e^{\frac{1}{z^2}}}{z^2+1}$

Compute the integral using residues: $\int_{|z-i|=\frac{3}{2}}\frac{e^{\frac{1}{z^2}}}{z^2+1}$ Inside the circumference there are the following singular points $-i$ which is a pole of order 1 and $0$ ...
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$f\left( z\right) =\dfrac {e^{-z}}{\left( z-1\right) \left( z+2\right) ^{2}}$ function $0 <\left| z+2\right| <3$ open the Laurent series in the area

$f\left( z\right) =\dfrac {e^{-z}}{\left( z-1\right) \left( z+2\right) ^{2}}$ function $0 <\left| z+2\right| <3$ open the Laurent series in the area How do I edit a function?
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Annulus of convergence for laurent series

I have the Laurent series $$\sum_{n=1}^\infty \frac{n! z^n}{n^n} + \sum_{n=1}^\infty 2^n/z^n $$ and want to determine the annulus of convergence. The second sum clearly converges when |z|>2, but I ...
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Singular part of $f(z) = \frac{z}{(2 + \log z)^{2}}$

I'm asked to find the "singular part" of $f(z) := \frac{z}{(2 + \log z)^{2}}$ (I think this means the principal part of the Laurent series -- this question is from an older qualifying exam at my ...
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Laurent Expansion $\frac{1}{z(1-z)^2}$

Hi I was wondering if anyone could help me with this Laurent expansion $ f(z)=\frac{1}{z(1-z)^2} $ about $z=1$ I don't think I have done it correctly but this is what I did: $f(z)=\frac{1}{z} \frac{...
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Prove that the complex function: $\frac{z}{ \sin(\pi/z)}$ has an anti-derivative on $\mathbb{C} \setminus D(0,1)$

Prove that the function $\frac{z}{\sin(\pi/z)}$ of a complex variable has an anti-derivative on $\mathbb{C} \setminus D(0,1)$. My attempt: I tried to develop the Laurent series at $z=0$ but without ...
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Why does $1/\text{series}$ equal $\text{series with changed signs}$?

Consider the following function and series: $$ f(z)=-\frac{1}{z} \frac{1-\frac{z^2}{2!}+O(z^4)}{1-\frac{z^2}{3!}+O(z^4)} \tag{1}.$$ I've seen on many posts here and in textbooks that the following ...
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Laurent series at infinity for $f(x) = x\arctan(x)$

How would I go about expanding this expression $$ f(x) = x\arctan(x)$$ into Laurents series at $x=\infty$. Substituting $y=\frac{1}{x}$ does not help me here, or I just do not understand how it would ...
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Have I found the correct Laurent series expansion?

I am supposed to find the Laurent series expansion for $$f(z) = \frac{z+1}{z^2-4}$$ in the region $1<|z+1|<3$. My solution: $$w=z+1 \Leftrightarrow z=w-1 \Rightarrow f(z) = \frac{w-1+1}{(w-1)...
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Residues for a Laurent series with two centres

I have worked out that the Laurent series for $\left(z^4sin(\frac{1}{z}) + (z+1)^4sin(\frac{1}{z+1})\right)$ is given by: $$\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!} \left( \frac{1}{z^{2n-3}} + \frac{...
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Finding a 0 centered Laurent Series

Stuck on trying to find the Laurent series for $$\frac{e^z -1}{z^2}$$ centered at $z_0 = 0$. Still new to Laurent series, so not entirely sure how to get it. I know the Taylor series for $e^z$ but don'...
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Singularity and Laurent series of several functions

For each of the following functions classify the isolated singularity at 0 and specify the principal part of the Laurent development there: a) $\dfrac{sin(z)}{z^n},\;n\in\mathbb{N}$ b) $\dfrac{z}{(...
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Laurent series of $f(z)=\frac{4z-z^2}{(z^2-4)(z+1)}$ in different annulus

Given $f(z)=\dfrac{4z-z^2}{(z^2-4)(z+1)}$ I need to find the Laurent series in the annulus: $A_{1,2}(0),\;A_{2,\infty}(0),\;A_{0,1}(-1)$ I found the following partial fractions: $f(z)=\dfrac{-3}{(z+...
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Is this a correct Laurent series expansion for the given annulus?

Expand the function $$f(z) = \frac{1}{(z + 1)(z + 3)}$$ in a Laurent series valid for $1 < |z| < 3$ My attempt: $$\frac{1}{(z + 1)(z + 3)}=\frac{1}{4}.\frac{1}{1+z}-\frac{1}{4}.\frac{1}{3+z}$$ ...
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Convergent region pf a Laurent series

How do I find the region of convergence in the following case: Let $f(z)=\sum_{i=0}^\infty a_iz^i$ be a power series with a positive radius of convergence. Determine the region of convergence of the ...
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Is there any method can find the coefficient of principal part and analytic part of the Laurent series(the product of two power series)?

For example: The Laurent series I can get the Laurent series by multiplying two power series. I can find coefficients by expanding these two power series and multiplying them one by one. but it takes ...
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Computing Pole Order

Determine the type of singularity of $z_0$ of $f(z)=\frac{1}{z-\sin(z)}\:z_0=0$. $\lim_{z\to 0}\frac{1}{z-\sin(z)}=\infty$ so it is clearly pole. I do not know if there is a way to compute the ...
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Polo Singularity and Laurent series

Let $z_0$ be a polo singularity, f(z) is analytic in the neighbourhood excluding $z_0$. Then $\phi(z)=\frac{1}{f(z)}$ which implies $\lim_{z\to z_0}\phi(z)=0$ So the $\phi(z)$ has the Laurent series:...
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Understanding the Laurent series $\frac{1}{a_n+a_{n+1}(z-z_0)+…}=c_{-n}+c_{-n+1}(z-z_0)+…$

Let $z_0$ be a polo singularity, f(z) is analytic in the neighbourhood excluding $z_0$. Then $\phi(z)=\frac{1}{f(z)}$ which implies $\lim_{z\to z_0}\phi(z)=0$ So the $\phi(z)$ has the Laurent series:...
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Finding $f(z)=\log(\frac{(z+1)^2}{(z^2+4)})$ Laurent series

Determine the Laurent series of the function $f(z)=\log(\frac{(z+1)^2}{(z^2+4)})$ on the set $A=\{z|\:2<|z|\}$. I thought of using the following expansion: $\log(1+z)=\sum_\limits{n=1}^{\infty}\...
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$f(z)=\frac{1}{z^2+1}$ Laurent Series

Determine the Laurent series of the function $f(z)=\frac{1}{z^2+1}$ on the set $A=\{z|\:0<|z-i|<2\}$. I know I should use the expansion $\frac{1}{z-1}=\sum_\limits{n=0}^{\infty}z^n$ or $\frac{1}...
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Laurent series representation of given function

I wanted to find Laurent series representation of function $1/(e^{z} -1)$. So I took minus common and apply the series formula of $1/(1-z)$ and then I use series formula for each $e^{zn}$. But I am ...
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Find the Laurent series expansion of the following function and hence find the integral over the unit circle centred at i

We are going to be examined on using the Laurent series expansion to find integrals along simple closed curves. But the notes and lectures barely covered it and we only have 2 examples given. Can ...
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Residue at infinity of $(e^{2\pi z}-1)/(z(z^2+1)^2)$

I have trouble to proove that the residue at inifinity of this function is zero (I found that the sum of the residues at finite is zero). I tried to expand in series at $z=0$ the function $f(1/w^2)/w^...
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Laurent Series of rational function in z

Im attempting to solve a problem of defining the types of singularities of a complex function. I found that if the function has a Laurent series expansion with a finite number of terms in the ...
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understanding the difference between Laurent and Taylor series.

In my homework, I have a problem that says, Set $f(z)$ = $\frac{e^{z^2}}{z^4}$. $(a):$ Find the Laurent series for $f$ centered at $z_0 = 0$ $(b):$ Let $C$ be the positively oriented unit circle. ...
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Expand the Laurent series in the following regions

Expand $f(z) = \frac{1}{z(z-1)(z-2)}$ in the region $0 < |z| < 1$: Using partial fraction decomposition, $f(z) = \frac{1}{2} \cdot \frac{1}{z}-\frac{1}{z-1} + \frac{1}{2}\cdot\frac{1}{z-2}$. ...
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Expand $f(z)=\frac{1}{z(z-3)}$ as laurent series in domain $1 < |z-4| < 4$

Expand $f(z)=\frac{1}{z(z-3)}$ as laurent series in domain $1 < |z-4| < 4$ Any suggestion i have $\frac{1}{z-3}= \sum_{n=0}^{\infty} (-1)^n \frac{1}{(z-4)^{n+1}}$ and $\frac{4}{z}= \...
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Laurent series of $\sin z/(1 - \cos z)$

I have trouble solving this exercise: find the first three terms of the Laurent series of $\sin z/(1 - \cos z)$ centered at $z=0$. I have found the first two. I proved that at $z=0$ we have a first ...
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Fourier coefficients of $f\left(z\right)=\frac{1}{1+\cos z}$ through Laurent series

I had to find the Fourier coefficients of this simply periodic function $$f\left(z\right)=\frac{1}{1+\cos (z)},$$ I proceeded considering the $w=exp(iz)$ and considering the Laurent expansion of the ...
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Finding the order of the pole of the complex function $f(z)=\frac{1}{\cos(z)-\sin(z)}$

I am new to complex analysis , this was a example problem and the author just says that as $z=\pi/4$ an isolated singularity , it is clear that the order of the pole is one. But I am not able to see ...
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Find $\oint\exp((z-1)^{-1})(z+3)^{-1}\,dz$

Compute $\oint\exp((z-1)^{-1})(z+3)^{-1}\,dz$ I have to calculate this. On the contour : $|z+3| = 7$. First I want to use the residues formula for this, so I calculated the residue at the simple pole ...
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Why do these laurent series approaches conflict?

I was working on a problem of finding the Laurent series of $\frac{1}{z-3}$ that converges where $|z-4| > 1$ So I had one approach, let $u=z-4$ then: $$\frac{1}{z-3} = \frac{1}{1+u} $$ $$ = \...
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On singularity of limit holomorphic function

Let $P\in \mathbb C$ and $f_n : D(P,r)\setminus \{P\} \to \mathbb C$ be a sequence of holomorphic functions such that there is a holomorphic function $f: D(P,r)\setminus \{P\} \to \mathbb C$ such that ...
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What type of singularity is $z=\infty$ for $f(z)=\frac{1}{(sin(1/z))}$?

Consider the function $$f(z)=\frac{1}{(sin(1/z))}$$ At $z=\infty$ does $f$ have an isolated singularity or not? Or is $z=\infty$ a regular point? $f(1/t)=1/(sin(t))$ has simple poles in $t=k \pi$, ...
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What is the Puiseux series of the Bessel function $J_n(n)$?

By numerical experimentation I find the first three terms of the Puiseux series of the Bessel function of the first kind $$ J_n(n) = \frac{\Gamma(\frac13)}{2^{2/3}\cdot 3^{1/6} \cdot \pi}n^{-1/3} - \...
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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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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 $...