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.
1,790
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Laurent series for a polynomial in the numerator
I have been given the following function for which I am supposed to construct the laurent series around the singular point -1:
$$f(z)=\frac{z^2+4z+4}{z+1}$$
From the general formula I know that this ...
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Did I do this Laurent series correctly?
I am wondering whether I did the Laurent series for the following function correctly since I am pretty unsure as to what I am doing there:
$$f(z)=-\frac{6}{(z-2)(z-4)}$$ for $2<|z|<4$.
Through ...
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2
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Identical Functions in a Domain
Problem statement:
If f1(z) and f2(z) are analytic in a domain D and equal at a sequence of points zn in D that converges in D, show that f1(z)=f2(z).
I'm not clear on this - I understand that since ...
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Laurent series of $f(z)=\frac {\exp (1/z)}{1-\sin(1/z)}$ around 0.
as you can see i would like to expand in a Taylor-Laurent around 0 the function in the title. I can kinda see that it should be an essential singularity, because it should be for ${\exp (1/z)}$ and ...
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Laurent series expansion in 2 complex variables
I'm reading about the $j$-function from Washington's book. We define $S_N$ to be the set of matrices of the form $\begin{bmatrix}a & b\\ 0 & d\end{bmatrix}$ where $a,b,d\in\mathbb{Z}, ad=N$ ...
- 510
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Laurent series of $\ln(1-\frac{2i}{z+i})$
How to find Laurent series of function $ln\left(1-\frac{2i}{z+i}\right)$ if it is known $1 \leq |z| \leq 2$.
I am confused how to find this, because I tried to prove that $|\frac{2i}{z+i}|<1$ and ...
- 429
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Residue of a Dirichlet Series at $s=1$
I have encountered this problem of determining the leading term in the Laurent expansion of a Dirichlet series. Let $d(n)$ be integers and consider the Dirichlet series $$D(s)=\sum_{n=1}^{\infty}\frac{...
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Equivalence between removable singularity criteria
I have observed, from my complex analysis course and Wikipedia respectively, that a function $f: \mathbb{C} \to \mathbb{C}, \, z \mapsto f(z)$ having a removable singularity at $z_0$ is equivalent to ...
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Expansion of Beta function near negative integers
Is there a closed form expression for the coefficients (after Taylor expansion) of the Beta function $B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$ around an arbitrary negative integer let's say $x=-...
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Laurent series of the function $f(z)=\frac{1}{z^2(e^{z}-e^{-z})}$
Find the first three nonzero terms of the Laurent series of $f(z)=\frac{1}{z^2(e^{z}-e^{-z})}$ in the annulus $0 \leq \lvert z \rvert \leq \pi$.
Now, instead of $f(z)$ I found the Taylor series of its ...
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Find the laurent series of the function $f(z) = \frac{1}{(z+i)(z-5i)}$ which is valid in the annulus where $1 < |z| < 5$
Find the laurent series of the function $$f(z) = \frac{1}{(z+i)(z-5i)}$$ which is valid in the annulus where $1 < |z| < 5$
This is where I'm at currently. I'm not sure if this is the correct ...
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Laurent Expansion at specific singularity
The main part of the Laurent expansion with regard to the singularity z=$-\pi$ is to be given by the following:
$$\frac{cos(z)}{(z^{2} - \pi^{2})(z+ \pi ) }$$
It is clear to me that it is a pole of ...
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Contour Integral around the unit circle $C$: $\oint_C \frac{e^z-1}{\sin^3(z)}dz$
Studying once again for my last attempt at the complex analysis qualifying exam. I'm a bit confused as to what to do with this contour integral, where $C$ is the unit circle.
$$\oint_C \frac{e^z-1}{\...
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Poles and the Laurent series for $\frac{1}{z^4-1}$
In a problem, I have been asked to find the poles of the following function:
$$f(z) = \frac{1}{z^4-1}$$
Of course, $f(z) = \frac{1}{(z+1)(z-1)(z+i)(z-i)}$. So the singularities are $z=\pm1$ and $z=\pm ...
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Find coefficients of Laurentseries
I find myself struggling a bit with Laurent series, and I'm currently stuck on this exercise below.
I want to find the coefficients $c_{-1}, c_0, c_1, c_2, c_3$ of the Laurent series of $\frac{1}{\sin(...
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Ring of formal multivariate Laurent series as a consecutive localisation
How do we localise a the ring of formal power series $\mathbb C[[x, y]]$ to get the field of multivariate Laurent series $\mathbb C((x))((y))$? If $S_x$ and $S_y$ are the multiplicative closed sets $\{...
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Decomposition of the field of Laurent formal power series in a polynomial ring and a formal power series ring
Can we decompose the field of formal Laurent series as
$$\mathbb C((t))\cong \mathbb C[t^{-1}]\times\mathbb C[[t]]$$
as vector spaces over the field of complex numbers? The map
$$\sum_{i\in\mathbb Z}\...
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Laurent series of real part of holomorphic function
In the book "Harmonic function theory" p. 9 there is this claim: let $f(z)=\sum_{j=0}^{+\infty}a_jz^j$ be a holomorphic function. Then its real part takes the form
\begin{equation}
u(r\xi)=\...
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What prerequisites do I need to understand the insertion of a lag-operator for a complex dummy z
In books on time series the following is frequently used:
We are given a power series $C(z):=\sum_i C_i z^i$, wherein the $C_i$ are matrices and the series is potentially infinite.
Subsequently a ...
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How can $1/f(z)$ have a zero?
On page 173 of Complex Analysis, Gamelin, the author states the following theorem:
Let $z_0$ be an isolated singularity of $f(z)$. Then $z_0$ is a pole of $f(z)$ of order $N$ if and only if $1/f(z)$ ...
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Clarification on Laurent series expansion exercise
I have two questions about my two different attempts at the following exercise.
Question 1: Does the first method = attempt 1 not work because we can concretely determine what each term of the Laurent ...
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Integral using Bessel functions $\int_0^{\pi}\cos(z\sin x)e^{\cos (x)}\ dx$
I want to evaluate the following,
$$I(z)=\int_0^{\pi}\cos(z\sin x)e^{\cos (x)}\ dx,\quad z\in\mathbb{N}$$
using Bessel functions. My attempt where I ended up with a divergent series is below. I think ...
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Showing holomorphicity of $f(z)=\sum_{n\ge 1} \frac{(z-n)^{-n}}{n!}$ on $\mathbb{C}\setminus \mathbb{N}$
I would like to construct a function that is holomorphic on $\mathbb{C}\setminus \mathbb{N}$ with a pole of order $n$ at each positive integer $n$. My idea was to take the following $f$:
$$f(z)=\sum_{...
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Determining the residue of a function by computing its laurent expansion
I need to find the residue of $$f(z) = \frac{e^z\sin(z)}{z(1-\cos(z))}$$ at $z = 0$ via its Laurent series expansion. First of all, I tried to expand all the functions via Taylor series at $z = 0$.
I ...
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Laurent expansion for $f(1/z)$, with $f$ an integer function.
The following is part of an exercise I am demonstrating, below I will ask my question.
I will detail a little of what I have:
Let $f$ be a holomorphic function on $\mathbb{C}$, $f$ is injective. Let's ...
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An easy way to calculate the residue of $\tan^2z$
I need to calculate the value of residue of $f(z) = \tan^2z$ at $z_k=\frac{\pi}{2} + \pi k$. I know that these points are poles of the second order, so the common way is to calculate residue by the ...
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Laurent series for $\frac{z}{1+z^3}$ for $|z|>1$
I managed a decent start but got stuck:
$|z|>1$ implies $1/|z|<1$, so we can use $\sum_{k=0}^{\infty}z^{-k} = \frac{1}{1-1/z}$ to help find the Laurent series:
$$\frac{z}{1+z^3} =\frac{z-1}{1+z^...
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Laurent series expansion in a region
Consider the function $$f(z) = \frac{1}{z(z-1)(z-2)}$$ in the region $2 < |z| < \infty$. I would like to find the partial fraction decomposition and the Laurent series expansion of $f(z) $ in ...
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Residues at Infinity $\oint_C \frac{\sin ^2 z}{z^2(z-1)}dz$
Evaluate the integral $$\oint_C \frac{\sin ^2 z}{z^2(z-1)} \mathrm{d}
z$$ taken counterclockwise around circles $C:|z|=2$.
This is a relatively simple problem that can be solved using residue ...
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Idempotents in Laurent series
I think it’s true that if $R$ is any commutative, unital ring, then if the formal Laurent series ring $R((t))$ is isomorphic to a Cartesian product of two rings $R((t)) \cong S_1 \times S_2$, then ...
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Minimal field containing two field of formal Laurent series
Let $L/K$ be a field extension and $X,Y$ be two $K$-algebraically independent elements in $L$.
What is the minimal subfield of $L$ that contains both $K((X))$ and $K((Y))$?
The field $K((X,Y))=\...
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if $ϵ>0$ and $f(z) = z + \frac{d_1}{z} + \frac{d_2}{z^2} + \cdots$ for $|z| > 1$ univalent, there is $C$ with $d_n \leq C(1 + ϵ)^n$ for all $n$
Let $f : \{ |z| > 1\} \rightarrow \mathbb C$ be a univalent function with a Laurent series expansion
$$f(z) = z + \frac{d_1}{z} + \frac{d_2}{z^2} + \frac{d_3}{z^3} + \cdots.$$
Prove that, or all $\...
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Uniqueness of a Contour Integral
Is the result of a contour integral around a closed loop path unique? I'd think it'd be similar to a definite integral, which is unique.
This came up regarding the proof of uniqueness of a Laurent ...
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Laurent expansion around 0
Given the function $$f(z) = z^2 \cdot \frac{{e^{1/z}}}{{z-1}}$$, we want to find its Laurent series expansion around the point $z = 0$.
This is my approach. I am not sure about the ending.
First, let'...
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How to classify the singularity of $f$ using Laurent series by considering $g=1/f$
I have the following question, and I am struggling to interpret what it wants me to show.
Let $U$ be an open subset of the complex plane, let $a \in U$ and let
$U'= U \backslash \{a\}$. Assume that $...
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Principal part of Laurent series when there's multiple singularities
I'm taking a complex variable class and I've recently been introduced to the Laurent series as
$$f(z)=\sum_{-\infty}^{\infty}a_n(z-z_0)^n$$
where $z_0$ is the singularity point and the expression ...
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Solving Power Series Equations if we introduce Logarithmic Terms
If we have a complex power series equation like
$$
\sum_{n=0}^{\infty} a_n z^n = \sum_{n=0}^{\infty} b_n z^n
$$
then we can conclude $a_n = b_n$. We can see this by viewing $z^n$ as basis elements, or ...
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Determining type of singularities for $f(z) = \frac{z}{\sin(z)}$
I want to determine the singularities for:
$f(z) = \frac{z}{\sin(z)}$ for $z_0 = 2\pi k $, where $ k\in \mathbb{Z} $.
If the singularity is removable, I have to compute the limit.
If there is a pole, ...
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in the Laurent expansion formula, where is the domain of definition of the functions $g(z)$ and $h(z)$?
In my textbook, Laurent's theorem is presented in this way:
Let $f(z)$ be analytic in a domain containing two concentric circles $c_{1}$ and $c_{2}$with
center $z_{0}$ and the annulus between them (...
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Are all singularities of $\tan(z)$ simple poles?
$\tan(z)$ has singularities at points $z=(2n+1)\frac{\pi}2$, where $n$ is an integer.
Are all these points of singularity a simple pole? Or do we need to obtain the type of singularity at each point ...
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What is the Grothendieck (K_0) group of the ring of Laurent polynomials?
I am curious to know what is the Grothendieck group of the ring of Laurent polynomials over a field. I am a beginner in the study of Algebraic K-theory. I have learned that the Grothendieck group of ...
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Laurent series expansion of $\frac{1}{(z+1)(z+2)}$
Expand $f(z) = \frac{1}{(z+1)(z+2)}$ in the region $1<|z|<3$.
Here's my working: By partial fractions,
$f(z) = \frac{1}{z+1}-\frac{1}{z+2}=\frac{1}{z(1+1/z)}-\frac{1}{z(1+2/z)}$
Now I can ...
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How to express the derivative of a function as its integral?
In the Laurent series, $${\displaystyle f(z)=\sum _{n=-\infty }^{\infty }a_{n}(z-c)^{n},}$$ the coefficients are the integrals $${\displaystyle a_{n}={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {f(z)}{(...
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Residue of a pole, removable singularity
I'm uncertain about poles and removable singularities. The function $g(z) = \frac{2z}{(1-z^2)^2} = 2z \cdot \frac{1}{(1-z)^2} \cdot \frac{1}{(1+z)^2}$ has singular points at $z_1 = 1, z_2 = -1$.
...
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Laurent series of the inverse of a perturbed singular matrix
Let $M(\epsilon)$ be the square matrix $M(\epsilon) = A + \epsilon B$ where $A,B$ are square matrices. Throughout this post, suppose that $M(\epsilon)$ is invertible for arbitrarily small but positive ...
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Laurent Series of $\frac{1}{(z^2+1)(z-1)^2}$
Find the Laurent series of $\dfrac{1}{(z^2+1)(z-1)^2}$ and use this to identify singularities and their residues.
$$\dfrac{1}{(z^2+1)(z-1)^2}=\dfrac{i}{2}\dfrac{1}{z^2+1}+\dfrac{i-1}{2}\dfrac{1}{z-1}+\...
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Residues of Implicit Meromorphic Function
There are lots of questions on this site asking how to find a Laurent expansion for functions involving rational functions, exponential functions and some common friends. I have encountered a ...
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Using long division with negative exponents
I have the following function
$$f(x)=\frac{\sum_{n=0}^{+\infty}a_n \, x^{-n}}{\sum_{n=0}^{+\infty}\left(b_n \, x^n + c_n \, x^{-n}\right)}$$
Is there a way, maybe using long division to find the ...
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How to expand a series in points where the denominator vanishes?
When I ask Mathematica to compute the series of $f(x)=\frac{\cos(x)}{\sin(x)}$ around $0$, I get:
$$\frac{1}{x}-\frac{x}{3}-\frac{x^3}{45}-\frac{2 x^5}{945}-\frac{x^7}{4725}-\frac{2 x^9}{93555}+\dots\...
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Do formal Laurent series of norm 1 over a local field come from power series over ring of integers?
Let $K/\mathbb Q_p$ and $L/K$ be finite extensions of fields, with $O_K$ resp. $O_L$ being the rings of integers in $K$ resp. $L$. Let $N:L\to K$ denote the norm map from $L$ to $K$. Consider the ...
