All Questions
Tagged with complex-analysis power-series
160
questions
15
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About the limit of the coefficient ratio for a power series over complex numbers
This is my first question in mathSE, hope that it is suitable here!
I'm currently self-studying complex analysis using the book by Stein & Shakarchi, and this is one of the exercises (p.67, Q14) ...
2
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3
answers
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Difficulties performing Laurent Series expansions to determine Residues
The following problems are from Brown and Churchill's Complex Variables, 8ed.
From §71 concerning Residues and Poles, problem #1d:
Determine the residue at $z = 0$ of the function $$\frac{\cot(z)}{...
- 623
8
votes
4
answers
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Proof that Radius of Convergence Extend to Nearest Singularity
Can someone provide a proof for the fact that the radius of convergence of the power series of an analytic function is the distance to the nearest singularity? I've read the identity theorem, but I ...
user82004
4
votes
4
answers
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How do I obtain the Laurent series for $f(z)=\frac 1{\cos(z^4)-1}$ about $0$?
I know that $$\cos(z^4)-1=-\frac{z^8}{2!}+\frac{z^{16}}{4!}+...$$ but how do I take the reciprocal of this series (please do not use little-o notation)? Or are there better methods to obtain the ...
- 39.9k
7
votes
3
answers
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Two different expansions of $\frac{z}{1-z}$
This is exercise 21 of Chapter 1 from Stein and Shakarchi's Complex Analysis.
Show that for $|z|<1$ one has $$\frac{z}{1-z^2}+\frac{z^2}{1-z^4}+\cdots +\frac{z^{2^n}}{1-z^{2^{n+1}}}+\cdots =\frac{...
- 5,601
13
votes
1
answer
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Radius of convergence of power series
Given a meromorphic function on $\mathbb{C}$, is the radius of convergence in a regular point exactly the distance to the closest pole?
As Robert Israel points out in his answer, that this is of ...
- 4,681
4
votes
2
answers
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Let $f$ be an analytic isomorphism on the unit disc $D$, find the area of $f(D)$
Let $f$ have power series $f(z) = \sum_{n=1}^\infty a_n z^n$ in $D$, then prove that $\mathrm{area}\, f(D) = \sum_{n=1}^\infty n \,|a_n|^2$.
Note: We define $\mathrm{area}\, S = \iint_S \mathrm{d}x\...
- 6,159
3
votes
1
answer
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Riemann Zeta Function Manipulation
The Riemann zeta function is defined on the $Re z> 1$ by $$\zeta(z)=\sum_{n=1}^\infty \frac{1}{n^z}$$
(i) show that for $Re z> 1$, we have $$(1-2^{1-z})\zeta(z)=\sum_{n=1}^\infty \frac{(-1)^{n+...
- 193
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votes
3
answers
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What is the Riemann surface of $y=\sqrt{z+z^2+z^4+\cdots +z^{2^n}+\cdots}$?
The following appears as the second-to-last problem of Stewart's Complex Analysis:
Describe the Riemann surface of the function $y=\sqrt{z+z^2+z^4+\cdots +z^{2^n}+\cdots}$.
This problem ...
- 16.8k
4
votes
2
answers
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How to find the radius of convergence?
The function is $\dfrac {z-z^3}{\sin {\pi z}} $. How to find the radius of convergence in $ z=0 $?
- 1,133
4
votes
1
answer
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What are the subsets of the unit circle that can be the points in which a power series is convergent?
Let $A\subset\Bbb C$ be a subset of the unit circle. Consider the following condition on $A$.
Cond. There exists a sequence $\{a_i\}_{i=1}^\infty$ of complex numbers such that $$\sum_{n=1}^\infty ...
- 6,315
1
vote
5
answers
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Finding coefficients of laurent series for $\frac 1{1-\cos z}$ about zero
I know I'm wrong, but I fail to see why I'm wrong. My goal is to try and find the terms for the Laurent series of $f(z)=\frac{1}{1-\cos(z)}$ but I'm surely off.
$$\begin{align} f(z)&= \frac{1}{1-\...
- 408
0
votes
2
answers
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Behavior of $\sum_{n=1}^\infty n^{-1}z^n$ on the circle of convergence
Consider the following complex power series :$$\sum_{n=1}^\infty\frac{z^n}{n}$$ The radius of convergence of this series is $1$ and the series is divergent for $z=1$. I want to know what are the ...
- 10.2k
12
votes
3
answers
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Expressing the area of the image of a holomorphic function by the coefficients of its expansion
I have the following problem.
Let $f:D\to \mathbb C$ be a holomorphic function, where $D=\{z:|z|\leq 1\}.$ Let $$f(z)=\sum_{n=0}^\infty c_nz^n.$$ Let $l_2(A)$ denote the Lebesgue measure of a set $...
- 6,315
7
votes
2
answers
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What is the radius of convergence of $\sum z^{n!}$?
How to find the radius of convergence of $\sum z^{n!}$?
I'm used to applying the ratio test to power series of the form $\sum a_{n}z^{n}$, but for a different power of $z$, I am a bit stumped. What ...
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