All Questions
20
questions
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83
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Problem 6.2.10 From Complex Analysis by Jihuai Shi
Definition: let $f$ be holomorphic on a region $G$ (here region means a non-empty connected open set). For $\xi\in \partial G$, if there exists a ball $B(\xi,\delta)$ and a holomorphic function $g \in ...
- 11
0
votes
1
answer
99
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can we have a Taylor series expansion of the function $\frac{1}{\zeta(s)}$?
can we have a Taylor series expansion of the function $\frac{1}{\zeta(s)}$at $s=1$?
if so how can we determine the radius of convergence of this expansion without assuming the truth of the riemann ...
- 43
1
vote
0
answers
86
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How to show that $f^{\star}\in\mathcal{H}(G^{\star})$ where $f^{\star}(z)=\overline{f(\frac{1}{\overline z}) }$?
$G\subset \Bbb{C}\setminus \{0\}$ be a domain (open and connected). Let $G^{\star}$ be the reflection of $G$ w.r.to the unit circle $S^1$. Given $f\in\mathcal{H}(G) $ ,then show that $f^{\star}\in\...
- 13k
1
vote
2
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88
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How to show that $\sum_{n=1}^{+\infty}\frac{z^{2n}}n$ can be analytically continued for $|\Re z|<1$?
Let's consider the following power series: $\sum_{n=1}^{+\infty}\frac{z^{2n}}n$
I think that the convergence radius is $R=1$.
Then the function given by $f(z)=\sum_{n=1}^{+\infty}\frac{z^{2n}}n$ is ...
- 23
5
votes
1
answer
450
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Extending $\sum_{n=0}^\infty s^{n^2}$ beyond its natural boundary
Let $\mathbb{D} = \{s \in \mathbb{C} : |s| < 1\}$. Let $f : \mathbb{D} \rightarrow \mathbb{C}$ where
$$ f(s) = \sum_{n=0}^\infty s^{n^2} $$
$f$ is analytic on $\mathbb{D}$. This is what it looks ...
- 6,027
1
vote
1
answer
194
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Analytic continuation of power series of holomorphic with real nonnegative coefficients
Consider a holomorphic function $f(z)$ defined as power series $$f(z)=\sum_{n=0}^{\infty}a_{n}z^{n},$$ with radius of convergence $R=1$.
I want to show that
If $a_{n}\geq 0$ for all $n$, then $f(z)$ ...
- 5,930
1
vote
2
answers
834
views
Radius of a convergence of power series of holomorphic function about a point
I observed that the radius of convergence of power series in the complex plane is basically the distance to nearest singularity of the function.I want to know whether my observation is correct or not....
- 733
8
votes
1
answer
290
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$f(z) = z + f(z^2)$ outside the unit disk?
The function $$
f(z) = \sum_{n=0}^\infty z^{2^n}
$$
which satisfies the functional equation $f(z) = z + f(z^2)$ is a classic example of a function analytic in $\mathbb{D} = \{z:|z|<1\}$ that cannot ...
- 6,340
0
votes
0
answers
45
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Doubt regarding proving analyticity of a power series
I am self studying concepts of complex analysis from Ponnusamy and Silvermann "Complex Variables with Applications"
In the chapter of analytic continuation in basic concepts authors mention ...
user775699
1
vote
0
answers
53
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Analytic continuation of a class of functions
Let $\{m_i\}_{i=0}^\infty$ be a subsequence of nature number sequence and then consider the following function:
$$f(z)=\sum_{i=0}^\infty z^{m_i}, |z|<1,$$
which is an analytical function on $|z|<...
- 313
0
votes
1
answer
59
views
Is there a "monotonicity" property for analytic continuation?
If I have two complex functions defined by power series $A(z) = \sum a_n z^n $, $B(z) = \sum b_n z^n$ with $|a_n| \ge |b_n|$ for all $n$, and I know that $A$ converges in some set $U_1$ and defines a ...
- 137
0
votes
1
answer
122
views
Problem based on analytic continuation along a path
Can the function $f(z) = \sum_{n = 0}^\infty z^{n!}$ can be analytically extended outside the circle $|z|=1$
My attempt : I was trying using some some radius convergence and analytic continuation of ...
- 486
0
votes
0
answers
976
views
Analytic continuation along a curve
Exercise: Find the Taylor series of the function $\log(z)$ on the disk $|z-1|<1$ of his principal branch. Then, continue analytically the function along the curve $\gamma:z(t)=e^{it},\, 0\leq t \...
- 1,222
0
votes
1
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470
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show two analytic functions are not analytic continuations of each other
Show that the series
$\sum_{n=1}^{\infty} ({1\over 1 - z^{n+1}} - {1\over 1-z^{n}})$
represents two analytic functions in the regions $|z| \lt 1$ and $|z| \gt 1$, and these functions are not ...
- 37
1
vote
1
answer
313
views
Prove that the holomorphic function defined by a certain sum can't be extended to a neighborhood of 1
Problem:
Let
$$f(z) := \sum_{n=1}^\infty \frac{1}{n^4}z^{2^n}$$
noting that the sum has radius of convergence 1. Prove that there exists no neighbourhood U of 1 and holomorphic function g:$U\...
- 524