All Questions
18
questions
1
vote
1
answer
40
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Classifying the singularities of a function
I'm trying to classify the singular points of the following function but I'm having some troubles.
$$f(z) = \frac{z(z-1)^{1/3}\sin(1/z)}{(z^2+1)^3\sin(z)}$$
This is what I have so far:
$z=i, z=-i$ ...
- 531
0
votes
0
answers
34
views
Determining nature of complex singularities using Taylor and Laurent series
I'm facing the following function:
$$f(z) = \frac{z}{\sin z}$$
I know there are singularities at $z_k=k\pi$ with $k \in \mathbb{Z}$. Using the basical properties of the $\sin(z)$ function, I found:
$$\...
user1107523
1
vote
1
answer
46
views
How to prove or disprove a point is a singularity of an analytic function defined by a power series?
My question is: in general, how we can prove or disprove that a point is an singular point of a analytic function defined by a power series?
Since for a point on the circle of convergence of a power ...
- 53
1
vote
1
answer
106
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Questions concerning analytic and singularities of power series
In the book 'The Concrete Tetrahedron' by Manuel Kauers and Peter Paule on p30 reads or implies the function
$$f: C\backslash \{ ... ,-2\pi,-\pi,0,\pi,2\pi, ...\}\rightarrow C,\, f(z)=\frac z{\sin(z)}$...
- 415
0
votes
1
answer
370
views
If $f$ is entire and monotone on $\mathbb{R}$, then it's a polynomial or $f(1/z)$ has an essential singularity
In the answer to this post, the following fact seems implicitly used:
Suppose $f: \mathbb{C} \to \mathbb{C}$ is entire. Further, suppose $f$ is a bijection $\mathbb{R} \to \mathbb{R}$ and $z$ is real ...
2
votes
3
answers
91
views
Little question about holomorphic function.
I am working on a task where the following is given:
Let $r > 0$ and $f$ a holomorphic function on $U=B_r(a)$ so that $f(a)=0$ but $f'(a)\neq 0$.
The Solution claims the existence of a holomorphic ...
- 121
0
votes
0
answers
28
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Retrieving a singular pole from the quotient of the coefficients of the power series [duplicate]
I want to show the following statement:
Let $f$ be a meromorphic function on $\mathbb{C}$ with one and only one pole at $z_0\neq 0$. This pole can have any order $k>0$, but has to be the only pole ...
- 553
2
votes
2
answers
206
views
Series expansion of $\dfrac{z^2}{1+z^2}$
Consider the complex function $$f(z) = \dfrac{z^2}{1+z^2}$$
It has two isolated singularities at $z=\pm i$. So when we consider the series expansion at $z_0 = 0$, then the radius of convergence is $...
- 167
0
votes
1
answer
65
views
Power series representation of $\frac{1}{1-e^z}$ to classify singularity.
I want to classify the singularity of $\frac{1}{1-e^z}$ at $0$.
With a "direct" power series expansion, I got
$$\frac{1}{-\sum_{k=1}^\infty \frac{z^k}{k!}}$$
This looks like an essential ...
- 2,276
1
vote
0
answers
50
views
Prove that $z=0$ is an essential singularity of $g(z)=e^{\frac{1}{z}}f(z)$ when $f(z)$ is analytic on $z=0$
Let $f:\mathbb{C}\to\mathbb{C}$ be an analytic function on $z=0$ and let $g:\mathbb{C}\to\mathbb{C}$ such that $g(z)=e^{\frac{1}{z}}f(z)$. Given $f(z)\not\equiv0$, prove that $z=0$ is an essential ...
- 1,656
0
votes
1
answer
109
views
Singularity of $\sum a_n z^n$ at $z=R$
This is a theorem in the Bak-Newman book of Complez analysis. It is stated:
Theorem 1: If $\sum_{n=0}^\infty a_nz^n$ has radius of convergence $R<\infty$ and, for every $n$, $a_n$ is real and $\...
- 3,226
1
vote
2
answers
152
views
Does $\lim\limits_{z \to 0}\frac{e^{z^{-1}}}{\sin(z^{-1})}$ exist or not?
I used limit of the function at zero, and got that the limit is zero. So I said, while the limit existed and it is finite then the singularity is Removable Singularity. My function is $$f(z)=\frac{e^{...
- 319
0
votes
1
answer
93
views
Power series of a complex function about a removable singularity
Compute the first three terms in the power series of $f$ at 0.
$$
f(z) = \frac{1-cos(z)}{sin(z^2)}
$$
How should I do this since the function is undefined at $f(0)$?
I've seen some similar power ...
- 3
2
votes
0
answers
102
views
Series representation and isolated singularity
I am currently doing the following problem :---
Let $f$ be holomorphic on $\{z\in \Bbb C:|z|<2,z\not=1\}$. Let $f$
has an representation of the form
$f(z)=\sum_{n=0}^{\infty}a_nz^n,|z|<1$...
- 9,634
1
vote
1
answer
213
views
Condition for the power series to have a single singularity on its circle of convergence
I'm looking to the prove of the following statement:
If for the complex power series $\sum^\infty_{i=1} a_i z^i$ it holds that
$$ \limsup_{n \to \infty} \sqrt[n]{\left| \frac{a_n}{a_{n+1}} -z_0 \...
- 1,879