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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How to switch to a Laurent series' next convergence ring?

Given the Laurent series $\sum\limits_{k=-\infty}^\infty a_k^{(l)} z^k = f(z)|_{r_l<|z|<R_l}$ of a meromorphic function $f$ on $\mathbb C$ with convergence region $r_l< |z|< R_l$, one can ...
18
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1answer
896 views

An outrageous way to derive a Laurent series: why does this work?

I had to compute a series expansion of $1/(e^{x}-1)$ about $x=0$, and in the course of its derivation, I made a couple of manipulations that are not allowed mathematically. Still, comparing the final ...
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1answer
1k views

Why $e^{\pi}-\pi \approx 20$, and $e^{2\pi}-24 \approx 2^9$?

This was inspired by this post. Let $q = e^{2\pi\,i\tau}$. Then, $$\alpha(\tau) = \left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{24} = \frac{1}{q} - 24 + 276q - 2048q^2 + 11202q^3 - 49152q^4+ \cdots\...
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Fourier series is to Fourier transform what Laurent series is to …?

Since the coefficients $$a_k = \frac1{2\pi i}\oint_C\frac{f(z)}{(z-c)^{k+1}}\,dz$$ for the Laurent series $$f(z)\Big|_{r\le|z|\le R} = \sum_{k=-\infty}^{\infty}a_k\cdot(z-c)^k $$ of a function $f\...
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Type of singularity of $\log z$ at $z=0$

What type of singularity is $z=0$ for $\log z$ (any branch)? What is the Laurent series for $\log z$ centered at 0, if exist? If the Laurent series has the form $\sum_{k=-\infty}^{\infty} a_kx^k$, ...
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755 views

Limit approach to finding $1+2+3+4+\ldots$

When exploring the divergent series consisting of the sum of all natural numbers $$\sum_{k=1}^\infty k=1+2+3+4+\ldots$$ I came across the following identity involving a one-sided limit: $$\lim_{x\...
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How to find Laurent series Expansion

$f(z)$ is defined like this: $$ f(z) = \frac{z}{(z-1)(z-3)} $$ I need to find a series for $f(z)$ that involves positive and negative powers of $(z-1)$, which converges to $f(z)$ when $0 \leq |z - 1| \...
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2answers
18k views

Laurent-series expansion of $1/(e^z-1)$

Find the Laurent series for the given function about the indicated point. Also, give the residue of the function at the point. $$ z\mapsto\frac{1}{e^z - 1} $$ The point is $z_0=0$ (four terms of ...
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2answers
23k views

$e^{1/z}$ and Laurent expansion

$e^\frac1z$ is not holomorphic at $z=0$, but it is known that it can be expanded as $$e^\frac1z=1+\frac1z+\frac1{2!z^2}+\frac1{3!z^3}+\cdots$$ The coefficients of this Laurent expansion are computed ...
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603 views

The Laurent series of the digamma function at the negative integers

To find the Laurent series of $\psi(z)$ at $z= 0$, I would first find the Taylor series of $\psi(z+1)$ at $z=0$ and then use the functional equation of the digamma function. Specifically, $$\begin{...
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1answer
18k views

Difference between the Laurent and Taylor Series.

I have never taken complex analysis, but I am preparing for a GRE for this week end and I am trying to learn a bit about Laurent Series. So far what I get is that the Laurent Series are of form $$\...
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3answers
4k views

Prove $f$ entire and with a pole at infinity to be polynomial

The question has been asked before, but I can't seem to wrap my head around it. I guess I'm missing something. There is a certain part where I'm experiencing problems, and I can't seem to find an ...
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2answers
693 views

Laurent series for $f (z)=\frac {\sin (2 \pi z)}{z (z^2 + 1)}$

How Can find the Laurent series for this function valid for $0 <|z-i|<2$ $$f (z)=\frac {\sin (2 \pi z)}{z (z^2 +1)}$$ Let $g (z) = \sin (\pi z)$ $$\sin (\pi z ) = \sin( 2 \pi (z - i)) \cos (2 ...
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2answers
527 views

finding the expansion of $\arcsin(z)^2$

Is there a fast and nice way to find the expansion of $\arcsin(z)^2$ without squaring expansion of $\arcsin(z)$ ? For $|z|<1$ show that $$(\sin^{-1}(z))^2 = z^2 + \frac{2}{3}\cdot \frac{z^4}{...
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Prove that $\zeta(4)=\pi^4/90$

I am asked to "use the calculus of residues" to prove that $$\displaystyle\sum\limits_{n=1}^{\infty} \frac{1}{n^4}=\frac{\pi^4}{90}$$ I think I can do this given the Laurent series for $\cot z$ ...
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2answers
513 views

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

I'm really stuck on this, and I have no idea how to start. Writing it at $f(z)=e^z e^{\frac{1}{z}}$ and their expansions didn't really give any insight. I am aware it is possible to multiply the ...
7
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2answers
314 views

Expand: $e^{\sin z}$ at $z = 0$

I need hints to find the Laurent expansion of $\displaystyle e^{\sin z}$ at $z = 0$ in a simpler way. I am getting double series which I can't simplify. ADDED:: Can it be simpler that the below? I ...
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2answers
676 views

How to prove that field of rational functions is a *proper* subset of field of formal Laurent series? [duplicate]

Now, if $F$ is a field, I can prove easily that $F(x)\subseteq F((x))$ but I'm having problems to show this is a proper inclusion. If for example $F=\Bbb R$ or $\Bbb C$, I can take a well-known ...
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147 views

Evaluating $\oint_{|z|=2} \frac{\log{(z-\alpha)}}{z}-\frac{\log z}z \, \mathrm dz$

Let $\alpha \in (-1,0)$ and define $$f(z)= \frac{\log{(z-\alpha)}}{z}-\frac{\log z}z$$ where the logarithms take their principal values, i.e. the arguments of $z$ and $z-\alpha$ are between $-\pi$ and ...
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0answers
178 views

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 $Gl_n(A((t)))$,...
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2answers
2k views

Laurent series for $1/(e^z-1)$

Trying to compute the first five coefficients of the Laurent series for $$\frac{1}{e^z-1}$$ centered at the point $0$. I'm not seeing a way to use the geometric series due to the exponential. Any ...
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3answers
15k views

The degree of a polynomial which also has negative exponents.

In theory, we define the degree of a polynomial as the highest exponent it holds. However when there are negative and positive exponents are present in the function, I want to know the basis that we ...
6
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1answer
1k views

Understanding the proof of Laurent's theorem

I am reading the proof of Laurent's theorem from the book ''A first course in complex analysis with applications'' by Dennis G. Zill. Line 3 of the proof says the introduction of a crosscut between $...
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1k views

What is the Taylor series of $\frac{1}{\sin(z)}$ about $z_0 = 1$?

This was a exam question so I know it cannot take too long to write out the proof. Only I cannot see an answer. I would imagine you write $\sin(z) = \sin(1+(z-1)) = \sin(1)\cos(z-1) + \sin(z-1)\cos(1)...
6
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1answer
8k views

Laurent series for $\cot (z)$

I'm looking for clarification on how to compute a Laurent series for $\cot z$ I started by trying to find the $\frac{1}{\sin z}$. I've found multiple references that go from an Taylor expansion for $...
6
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1answer
131 views

When proving that f(z) is a polynomial, is it enough to consider just one point instead of keeping z arbitrary?

I think so - but I'd rather ask the MSE community too. Say I am given the bound |f(z)| < $|z|^3$, and that f is entire. Show f must be a polynomial. I used Cauchy's Integral Formula for ...
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1answer
174 views

Is there a completion of Laurent series w.r.t. integration?

Given a Laurent series $$f(z) = \sum_{k=-\infty}^\infty a_k(z-z_0)^k,$$ its derivative is obviously also a Laurent series. However, upon integration, the $\frac1{z-z_0}$ term introduces a $\ln(z-z_0)$ ...
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0answers
362 views

finding Laurent expansion of a periodic function

How are Laurent series and Fourier series related to each other? There is a problem that states that for a periodic function $F(z + 2 \pi ) = F(z)$ that is analytic in finite plane. $$F(z) = \sum_{n=-...
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3answers
15k views

How to find a Laurent Series for this function

How do I give a Laurent Series on various ranges of $|z|$? I need to find the Laurent series expansion for $$f(z)=\frac{1}{z(z-1)(z-2)}$$ for the following ranges of $|z|$: $0<|z|<1$ $1<|z|...
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3answers
271 views

If $f: \mathbb{C} \rightarrow \mathbb{C}$ is analytic and $\lim_{z \to \infty} f(z) = \infty$ show that $f$ is a polynomial

I'm learning about complex analysis and need some help with this problem: If $f: \mathbb{C} \rightarrow \mathbb{C}$ is analytic and $\lim_{z \to \infty} f(z) = \infty$ show that $f$ is a ...
5
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1answer
324 views

Pole of Riemann zeta and Riemann zeta zeros, prove this relation.

Prove this relation: $$\displaystyle \lim_{s\to 1} \, \left(\zeta (s)-\frac{\zeta '(s-1+\rho _n)}{\zeta \left(s-1+\rho _n\right)}\right)=\gamma -\frac{\zeta ''(\rho _n)}{2 \zeta '(\rho _n)}\;\;\;\;\;\...
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2answers
2k views

Calculating Laurent series expansion

I have to calculate the Laurent series expansion of $$f(z) = \frac {2z−2}{(z+1)(z−2)}$$ in $1 < |z| < 2$ and $|z| > 3$. For first annulus, I know I must manipulate the given expression ...
5
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2answers
835 views

Power series expansion of $\frac{z}{(z^3+1)^2}$ around $z=1$

I want to expand $f(z)=\frac{z}{(z^3+1)^2}$ around $z=1$. That is, I want to find the coefficients $c_n$ such that $f(z) = \sum_{n=0}^\infty c_n (z-1)^n$. So far, my first strategy was using long ...
5
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1answer
287 views

Laurent series, integral over the annulus, radii

We are given $$f = \sum_{n= - \infty} ^{\infty} a_n (z-z_0)^n \in \mathcal{O} ( \text{ann} (z_0, r, R)), \ \ 0<r<R< \infty. $$ Prove that $$\frac{1}{\pi} \int _{ann (z_0, r, R)} |f(z)|^2 d \...
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1answer
1k views

Laurent series of $\frac1{\sin^2z}$ around 0

I tried to expand $\frac{z^2}{\sin(z)^{2}}$ using Taylor expansion, but the coefficient involved some limit of $\frac{0}{0}$ and was really difficult to calculate. (I tried to convince myself the ...
5
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2answers
122 views

Laurent series of $z^{-3}$ at $z_0 = i$. Is there a way to do this by hand or is the question just evil?

I have to find the two Laurent series expansions of $\frac{1}{z^3}$ about $i$. The only approach I can think of is to do: $$\frac{1}{z^3} = \frac{1}{(z-i)^3} \left( \frac{z-i}{z} \right) ^3 = \frac{1}...
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1answer
3k views

Multiplying Laurent series

I was solving the following problem from Bak & Newmans Complex Analysis (Chp. 9 # 12b): Find the Laurent series (in powers of z) for $$\frac{1}{z(z-1)(z-2)}$$ in the open annulus $1 < |z| < ...
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2answers
139 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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2answers
2k views

How to obtain the Laruent expansion of gamma function around $z=0$?

I want to prove, the laurent expansion of gamma function. \begin{align} \Gamma(z) = \frac1z-\gamma+\frac12\left(\gamma^2+\frac {\pi^2}6\right)z-\frac16\left(\gamma^3+\frac {\gamma\pi^2}2+2 \zeta(3)\...
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1answer
1k views

Laurent series, radii of convergence.

I'm working on the following exercise: Prove that a Laurent series \begin{align*} \sum_{n = -\infty}^\infty a_n(z-z_0)^n = \sum_{n = 0}^\infty a_n(z-z_0)^n + \sum_{n = 1}^\infty a_{-n}(z-z_0)^{-n}...
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2answers
260 views

How to compute $\int_C {e^{3z}-z\over (z+1)^2z^2}$?

I am asked to compute the integral $$ \int_C {e^{3z}-z\over (z+1)^2z^2} $$ where $C$ is a circle with the center at the origin and radius ${1 \over 2}$. My approach was to separate the integral as a ...
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Algebraic elements in a bijection between $p$-adic numbers and formal Laurent series over $\mathbb F_p$

Let $p$ be a prime number. We have a bijection $$\begin{align*}\sigma : \mathbb Q_p & \to \mathbb F_p(\!(t)\!) \\ \sum_{i \geq -n} a_i p^i & \mapsto \sum_{i \geq -n} a_i t^i \end{align*}$$ ...
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91 views

Find the value of $a_{-2}$ in the Laurent series expansion of $\sin{(\frac{z}{z+1})}$

If $\sum_{-\infty}^{\infty}a_n (z+1)^n$ is the Laurent series expansion of $f(z)=\sin{(\frac{z}{z+1})}$, then find the value of $a_{-2}$. My work: So we are asked to find the laurent series ...
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241 views

Manipulation of Taylor/Laurent series

I have a question regarding how to expand a given rational function into its Taylor/Laurent series representation. Suppose we are given the function $$f(z) = \frac{z}{(z-1)(z-3)},$$ and are asked to ...
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0answers
154 views

Why is $a_{n}(x,y)=a_{n}(y)$?

This particular question is connected (with a slight variation in the definition of $g$) to an earlier question. The link is here. The specifics are: Given that $u(x,y)$ is the solution of a PDE ($x$ ...
4
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3answers
8k views

Calculate Laurent series for $1/ \sin(z)$

How can calculate Laurent series for $$f(z)=1/ \sin(z) $$ ?? I searched for it and found only the final result, is there a simple way to explain it ?
4
votes
3answers
6k views

How to find the Laurent expansion for $1/\cos(z)$

How to find the Laurent series for $1/\cos(z)$ in terms of $(z-\frac{\pi}{2})$ for all $z$ such that $0<|z-\frac{\pi}{2}|<1$
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2answers
2k views

Laurent Series at Infinity

I thought that finding the Laurent series was something that was straightforward, however, I am having some difficulty of finding the Laurent series of $$f(z) = \frac{1}{z(1-z)}$$ for $z= \infty$. ...
4
votes
6answers
11k views

Approximate $\coth(x)$ around $x = 0$

I'm trying to approximate $\coth(x)$ around $x = 0$, up to say, third order in $x$. Now obviously a simple taylor expansion doesn't work, as it diverges around $x = 0$. I'm not quite sure how to ...
4
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2answers
3k views

Type of singularities of $\frac{z}{e^z-1}$

I don't really understand how one can find the type of singularities for a given function. Say if $$f(z) = \frac{z}{e^z-1}$$ then I know that the singularities are at $z = 2n\pi i$ However, how do I ...