All Questions
Tagged with singularity-theory laurent-series
16 questions
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107
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Laurent series and apparent essential singularity
After reading "Mathematics for physicists" by Susan M. Lea I encountered a subtlety that I can't turn my head around (p. 128). Consider function
$$f(z)=\frac{1}{z^2-1} = \frac{1}{2}\left[\...
- 113
0
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1
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1k
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Singularities at infinity and laurent series
For this problem, I have three questions:
1. Why do we calculate the residue at the infinity? Is it because all the four poles are in contained in the disk?
2. When calculating $f(1/z)$, don't we just ...
- 1,437
1
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2
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670
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What is the image near the essential singularity $z=0$ of $\cos(1/z)$?
Determine the image near the essential singularity $z=0$ of the function $\cos(1/z)$. i.e. if $f(z)=\cos(1/z)$, What is $f\left(\mathcal{B}_{\varepsilon}(0)\right)$ for $\varepsilon > 0$?
Remark: ...
- 3,126
1
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2
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Finding the order of pole of $f(z)=\frac{\sin z}{z-\pi}$
The problem is Kreyszig 10ed international edition : 16.2 #9.
What is the order of the pole at $z=\pi$ of the function $f(z)$ below?
$$f(z)=\frac{\sin z}{z-\pi}$$
I thought that it will be a simple ...
- 701
4
votes
2
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5k
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Meromorphic Function on Extended Plane
How do I prove that every meromorphic function on the extended plane is a rational function?
- 952
2
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1
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72
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Question about finding Laurent Series over closed region and classifying singularity
Represent $\sin(\pi x/(x+1))$ Laurent Series about the region $0<|x+1|<2$:
Its true that $$\sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$$
So the $$\sin(\pi x/(1+x))=\sum (-1)^{n-1} \frac{(\pi ...
- 51
0
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0
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52
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Laurent series about which point?
I have been given the question: "Find the laurent series of $1/(1-z^3)$". No other information.
The question was posed in the process of finding the series' residual and in the answer I can see that ...
- 277
2
votes
1
answer
548
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Decomposition of resolvent in projections
I am reading the book Perturbation theory for linear operators from Kato.
He defines (§5 Section 3) for an operator $T : X\to X$ on a finite Banach Space the resolvent as
$$ R(x) = (T- x)^{-1}.$$
...
- 3,769
1
vote
2
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131
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Singularities of $\frac{1}{e^{\frac{1}{z}}+2}$ and classification?
I'm considering the function
$$\frac{1}{e^{\frac{1}{z}}+2}$$
Clearly, it has a unique singularity at $z=0$. I feel like its an essential but I can't find the Laurent expansion or any other way of ...
- 123
0
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1
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286
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Negative index coefficients of Laurent series for 1/sin(z)
Given $f(z) = \dfrac{1}{\sin(z)}$
a) Give singularities
b) Determine coefficients $a_{-1}$ and $a_{-3}$ of the Laurent series
So I thought:
a) $n \pi$, where $n$ is an integer
b) $\dfrac{1}{\sin(...
- 267
2
votes
1
answer
82
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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 $\;\Omega=\left\{z\in \mathbb{C}:2\lt\left|z-i\right|\right\}$
$$\begin{eqnarray}f(z)={1 \...
- 882
3
votes
1
answer
472
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Laurent series of an analytic function divided by $z$
This is a probably basic question about Laurent series. Say $g(z)$ is an analytic function, that $g(0) = 0$, and $f(z) = g(z)/z$. My textbook says $z = 0$ is a removable singularity of $f(z)$.
A ...
- 536
2
votes
2
answers
723
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Are the convergence radii circles of a Laurent-series always caused by isolated singularities?
Laurent series
$$f(z) := \sum_{n=-\infty}^\infty a_n (z-c)^n$$
converge for $r<|z-c|<R$ where
$$r = \limsup_{n\to\infty}|a_{-n}|^{\frac1n},
\\\frac1R = \limsup_{n\to\infty}|a_n|^{\frac1n}.$$
...
- 6,749
26
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3
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29k
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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 $\sum_{k=-\infty}^{\infty} a_kx^k$, ...
- 2,355
2
votes
1
answer
1k
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How to describe the singularities of a function
This is a question of an old exam of my complex analysis course and although I think I understand what a singularity is, I oftentimes have troubles 'finding and describing' them properly. I looked at ...
- 467
2
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2
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2k
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Removable singularity and laurent series
How to show $z=\pm\pi$ is a removable singularity for $\frac1{\sin z}+\frac{2z}{z^2-\pi^2}$?
I tried to compute the Laurent series, specifically the coefficients for $1/z,1/z^2,...$ since if we can ...
- 2,355