Questions tagged [laurent-series]

This tag is for questions about finding a Laurent series of functions and their convergence. 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.

287 questions with no upvoted or accepted answers
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7
votes
0answers
189 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 $Gl_n(A((t)))$,...
6
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0answers
160 views

Generator of inertia group in function field extension

Can Someone help me solve the following problem? Let C((T)) be the field of formal Laurent series in the variable T over an algebraically closed field C of characteristic $0$. (1) Prove that ...
6
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0answers
373 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=-...
5
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0answers
93 views

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 $$\begin{align*}\sigma : \mathbb Q_p & \to \mathbb F_p(\!(t)\!) \\ \sum_{i \geq -n} a_i p^i & \mapsto \sum_{i \geq -n} a_i t^i \end{align*}$$ ...
5
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0answers
94 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 ...
5
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0answers
251 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 ...
5
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0answers
156 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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0answers
71 views

Compute Galois group of extension

Let $C$ be an algebraically closed field with characteristic $p > 2$ What is the Galois group of the splitting field for the polynomial $F(X,T) = X(X+1)^p(X-1)^p - T$ over the field of formal ...
4
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0answers
122 views

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 $z_{0}$ ...
4
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0answers
217 views

Laurent expansion for $\sqrt{z(z-1)}$

Let $f(z) = \sqrt{z(z-1)}$. The branch cut is the real interval $[0,1]$, and $f(z)>0$ for real $z$ that are greater than 1. I need to find the first few terms of the Laurent expansion of $f(z)$ for ...
4
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0answers
2k views

Prove Laurent Series Expansion is Unique

Suppose that $f$ is holomorphic on $A=\{r<|z|<R\}$, where $0\le r<R\le \infty$. Suppose that there are two series of complex numbers $(a_n)_{n\in{\mathbb Z}}$ and $(b_n)_{n\in\mathbb Z}$ such ...
4
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0answers
402 views

Given a meromorphic function (via its Laurent series), how to obtain the (Taylor series of the) two holomorphic functions it is the quotient of?

Since any meromorphic function $f:\mathbb C\to\mathbb C$ can be expressed as the quotient of two entire functions, i.e. $f(z) = \frac{n(z)}{d(z)}$ where the zeros of the denominator $d(z)$ are $f$'s ...
3
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0answers
89 views

Finding Galois group of function field extension

Let $C((T))$ be the field of formal Laurent series in the variable $T$ over an algebraically closed field $C$ of characteristic $0$. I need to find to the Galois group of the splitting field for the ...
3
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0answers
67 views

Laurent Series for $f(z)=\frac{2}{(z+1)^2}-\frac{5}{z-5}$

I am trying to find the Laurent series for the function $$f(z)=\frac{2}{(z+1)^2}-\frac{5}{z-5},$$ in powers of $(z-1)$ that converges when $z=4$. The series will converge when $z=4$ is in the region $...
3
votes
2answers
32 views

Finding the laurent series with an added imaginary component

Find the laurent series of $$f(z)=\frac{i}{z^2-iz+2}$$ with the values of the region $a,b$ in which it is valid $$ a<|z-1|<b $$ My attempt at a solution: I did a partial fraction expansion ...
3
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0answers
56 views

Doubt on Laurent series of $f(z) = 1/z(z-2)^2$

I've a doubt about this exercise Find the Laurent series of the function $$f(z) = \frac{1}{z(z-2)^2}$$ addend $z=0$ in the annular regions of interest The problem it's not totally about the how ...
3
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0answers
98 views

Brauer group of the field of Laurent series with coefficients in a finite field

In a course I attended at university, we calculated the Brauer group of $\mathbb{F}_q((t))$ with $q=p^n$ , $p$ prime number and we proved it was $\dfrac {\mathbb{Q}}{\mathbb{Z}}=Br(\bar{\mathbb{F}_q}((...
3
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0answers
48 views

What is known about the image of the Jones polynomial functor (in particular on the class of all knots)?

Question. What exactly is known about the image of the Jones-polynomial-function $V$: $\{$ all links $\}$ $\rightarrow$ $\mathbb{Z}[t^{-1/2},t^{1/2}]$? Are there references explicitly and ...
3
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1answer
75 views

How do I find Laurent Serie for $\frac{1}{z(z-1)(z-2)}$

Find Laurent Serie in all three ranges. What I did: $\frac{1}{z(z-1)(z-2)}=\frac{1}{2}\frac{1}{z}+\frac{1}{z-1}+\frac{1}{2}\frac{1}{z-2}$ $\frac{1}{z-1}= -\sum\limits_{n=0}^{\infty} z^n, |z|<1$ ...
3
votes
2answers
62 views

Can a Laurent series be found for $f(z)=\frac{1}{(z+1)(z+2)}$ in the region $0<|z+1|<2$?

I know that a Laurent series can be found for $\frac{1}{(z+1)(z+2)}$ in the region $0<|z+1|<1$, but can a Laurent series be found for $0<|z+1|<2$? I am confused because in the region $0&...
3
votes
0answers
338 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 $\cot\left(\frac{1}{...
3
votes
0answers
423 views

$f$ 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 ...
3
votes
2answers
107 views

Is there a simpler way to compute the residue of a function at a pole of order 3?

The function $$\frac {1}{z^2(e^{i2\pi z}-1)}$$ has a triple pole at z = 0. To compute the residue of f at z = 0, I can compute the Laurent expansion of f about z = 0, and then read off the ...
3
votes
0answers
50 views

Finding the coefficients of the Weirestrass $\wp$ function.

I am trying to find the coefficients of the $\wp$-function. Right now I have the Laurent series about the pole $ z = 0$: $$\wp(z) = \frac{c_{-n}}{z^n} + \cdots + \frac{c_{-1}}{z} + c_0 + c_1 z + \...
3
votes
0answers
485 views

Finding Laurent Series of a function

I've been assigned to write a computer program which then calculates the Laurent series of a function. Of course I'm familiar with the concept, but I've always calculated the Laurent series in an ad ...
3
votes
0answers
551 views

Prove that periodic analytic function can be written as $\sum_{-\infty}^{\infty} c_n e^{2\pi inz}$

This question involves the following homework problem: PROBLEM Suppose $f$ is analytic in the upper half plane and periodic of period 1. Show that $f$ has an extension of the form $$f(z)=\sum_{-\...
3
votes
1answer
323 views

Laurent series of an analytic function divided by $z$

This is a probably basic question about Laurent series. Say $g(z)$ is an analytic function, that $g(0) = 0$, and $f(z) = g(z)/z$. My textbook says $z = 0$ is a removable singularity of $f(z)$. A ...
2
votes
1answer
50 views

proving ring of convergence $fg=\sum\limits_{m,n=-\infty}^{\infty} a_n b_m (z-z_0)^{n+m}$

how to prove this I am stuck at this point $$fg=\sum_{m,n=-\infty}^{\infty} a_n b_m (z-z_0)^{n+m}=\sum_{i=-\infty}^{\infty} c_i(z-z_0)^{2n}$$ what to do next ? anyone please help me in proving this
2
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0answers
27 views

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: $$\frac{1}{(z + 1)(z + 3)}=\frac{1}{4}.\frac{1}{1+z}-\frac{1}{4}.\frac{1}{3+z}$$ ...
2
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0answers
137 views

Explicit calculation of Laurent series coefficients by evaluating integral

I want the Laurent expansion \begin{equation} f(z) = \sum_{n=-\infty}^{\infty}c_n (z-a)^n \end{equation} of $f(z) = \frac{1}{(z+1)(z+3)}$ around the isolated pole $a =-1$ (for $|z+1|<2$). From a ...
2
votes
0answers
26 views

Is $G(k(t))$ dense in $G(k((t)))$ for $G$ a linear algebraic group?

Let $G$ be a linear algebraic group. Let $k$ be a field, let $k(t)$ be the field of fractions of the polynomial algebra $k[t]$ and let $k((t))$ be the field of Laurent series. I'm interested in ...
2
votes
1answer
49 views

Expand $z^4\cos(z-1)$ around $z=1$

Expand $z^4\cos(z-1)$ around $z=1$ to Laurent series We take $w=z-1$ $$(w+1)^4\cos(w)=(w+1)^4\sum_{n=0}^{\infty}(-1)^n\frac{w^{2n}}{2n!}=(w^4+4w^3+6w^2+4w+1)\sum_{n=0}^{\infty}(-1)^n\frac{w^{2n}}...
2
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0answers
131 views

Inverse Laplace Transform of a fractional form involving Modified Bessel Functions

I'm working on this inverse Laplace Transform problem $$ \mathscr{L^{-1}}\left(\frac{(BI_{0}(\surd{s}) - \frac{CI_{1}(\surd{s})}{\surd{s}}}{(I_{0}(\surd{s}) - \frac{AI_{1}(\surd{s})}{\surd{s}}}\right)...
2
votes
0answers
43 views

Solution to an $n^{th}$ order polynomial equation as a series in $n$

If I have an $n^{th}$ order polynomial set equal to zero, is there some way I can invert it to get a series expansion in terms of $n$? In particular I'm interested in solving the equation $$ U_n(x) -...
2
votes
0answers
137 views

Laurent series of $ \frac{z-12}{z^2 + z - 6}$ valid for $|z-1| >1$

Find the Laurent series for $f(z) = \frac{z-12}{z^2 + z - 6}$ valid for $(a) \ \ \ 1 < |z-1| <4$ $(b) \ \ \ |z-1| > 1$ $(c) \ \ \ |z-1| < 4$ We have $$f(z)= f(z) = \frac{z-12}{z^2 + ...
2
votes
1answer
82 views

Identity involving $\log$ and a power series

I'm trying to show that $$\log(z-z') = \log(-(z'-A)) - \sum_{n=1}^\infty \frac 1 n \left(\frac{z-A}{z'-A}\right)^n $$ where $A$ is a constant, and $\left|\dfrac{z-A}{z'-A}\right| < 1$. If it ...
2
votes
0answers
127 views

Dealing with a double summation series

Good morning, Is there a systematic way to deal with double series? I will give to you the specific series I would like to sum symbolically but I will be also interested in some reference where I can ...
2
votes
0answers
294 views

Laurent Series of $\frac{1}{(z-2)(z+1)}$

$$\frac{1}{(z-2)(z+1)}=\frac{A}{z-2}+\frac{B}{z+1}$$ \begin{cases}A+B=0 \\ A-2B=1 \end{cases} $$A=\frac{1}{3} \\ B=-\frac{1}{3}$$ $$\frac{1}{(z-2)(z+1)}=\frac{\frac{1}{3}}{z-2}+\frac{-\frac{1}{3}}{...
2
votes
0answers
42 views

How to find the constant $c_0$ term in $\displaystyle F(z) = \sum_{n=-Large1}^{Large2} {c_nz^{2n}}$

How to determine if $c_0 = 0$ ? Where $$F(z) = \left( \prod_{k = 0}^{N} (1+z^{2a_k}) \right) \left( \prod_{j = 0}^{M} (z^{-2b_j} + z^{-2d_j}) \right) = \sum_{n=-Large1}^{Large2} {c_nz^{2n}}$$ ...
2
votes
0answers
234 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)=\...
2
votes
0answers
91 views

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 ...
2
votes
1answer
48 views

Laurent series expansion of $f(z)$ which converges for $\frac12 <|z|<1$

My task is to find the Laurent series expansion of $$f(z)=\frac{1}{2z^2-z-1},$$ which converges for $\displaystyle \frac12 <|z|<1$. I proceeded by doing the following: $$\frac{1}{2z^2-z-1} = \...
2
votes
2answers
25 views

Find the Laurent series for $p(4/z)$

Find the Laurent series for $p(4/z)$ given that $p(z)=(z-3)^3$ My attempt: if the Taylor series for $p(z)$ looks like $$\frac{-27}{0!}+ \frac{27z}{1!}-\frac{18z^2}{2!}+ \frac{6z^3}{3!}+0+0+0...$$ ...
2
votes
0answers
38 views

Expanding $f\left ( z \right )=\frac{e^{az}}{1+e^{z}}$ about $z= i\pi$

$$f\left ( z \right )=\frac{e^{az}}{1+e^{z}} ,\left ( a\in\left ( 0,1 \right ) \right )$$ The point $z=i\pi$ is one of the nonremovable singularities of this function. In order to expand it about that ...
2
votes
0answers
27 views

Limit of $\frac{1}{2^n}\sum_{n=0}^{2^n-1}f(e^\frac{2k\pi}{2^n}i)$ for a complex-analytic function

Let consider a Laurent series $\displaystyle{ \sum_{k\in\mathbb Z}a_kz^k }$ with complex coefficients and converging inside the annulus $A=\{\ z\in\mathbb C\ |\ r<|z|<R\ \}$, with $r<1<R$....
2
votes
0answers
26 views

How come the definition of analytic continuation doesn't require the smaller and the bigger open subsets to be connected?

The reason that is making me think that these subsets should be connected / simpled connected is because I think that the Taylor disks of convergence of f and F, which is the continuation of f to the ...
2
votes
0answers
62 views

Laurent polynomial regression?

Polynomial regression is a common way of doing curvilinear regression. It is common to also use the inverse transform x^-1 (http://pareonline.net/getvn.asp?v=8&n=6). One can extend the concept ...
2
votes
0answers
217 views

Finding the Laurent series given the poles and residues

I am working on the following problem, suppose that $f$ has a simple pole at $-1$ with $Res(f,-1) = 1$. A double pole at $2$ with $Res(f, 2) = 2$. Also $f(0) = 7/4$ and $f(1) = 5/2$. I am supposed ...
2
votes
1answer
366 views

Uniform convergence in the Laurent coefficients proof, why?

Using the Laurent-expansion of $f(z)$ around $0$ $$f(z) = \sum_{n=0}^{+\infty} a_n z^n + \sum_{n=1}^{+\infty} b_n z^{-n} \tag{1}$$ Theorem The coefficients of the Laurent series are given by ...
2
votes
1answer
77 views

Question regarding singularity of a complex function

Consider the function $$f(z) = {1 \over (z-i)(z+i)}$$ with a Laurent series expansion at $z_0=i$ on a domain $\;\Omega=\left\{z\in \mathbb{C}:2\lt\left|z-i\right|\right\}$ $$\begin{eqnarray}f(z)={1 \...