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46 views

On sequences of rational functions [duplicate]

Let $\{f_n\}_{n=0}^\infty$ be a sequence of rational functions of the following form: $$ f_n(z) = \sum_{m=1}^\infty \frac{C_{m,n}}{z-m}$$ with $C_{m,n} \in \mathbb{Z}$, $C_{1,n} = 1$, and for each $n ...
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0answers
54 views

Analytic continuation of an integral

Let $$f(y)=\frac{y_1^{1/3}y_2^{1/3}}{y_1+y_2+1}$$. Consider the following integral: $$F(s_1,s_2)=\int_{\mathbb{R}_+^2}f(y)^{s_1}f(y^{-1})^{s_2}\frac{dy_1dy_2}{y_1y_2}$$ where ...
11
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1answer
356 views

Distribution of zeroes of lacunary functions

In a recent Math Stack Exchange question I asked about the function $$f(z)=\sum_{n=0}^\infty z^{2^n},$$ and was informed of its status is a canonical example of a lacunary series with natural boundary ...
5
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2answers
209 views

Simultaneous analytic continuation of Dirichlet eigenfunctions

Let $D\subset\mathbb{R}^d$ be a bounded domain which is regular for the Dirichlet problem. There is then a complete set of orthonormal eigenfunctions $\phi_j$ with corresponding eigenvalues ...
3
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2answers
462 views

Good book on analytic continuation?

This is a cross-post from MSE. For my Bachelor's thesis, I am investigating divergent series summation methods. One of those is analytic continuation. There are quite a few books on complex analysis ...
4
votes
2answers
492 views

I don't understand behavior of this integral, help!

In an answer to a question I needed the following integral: $$ f(z):=\int\limits_0^\infty t\coth(zt)e^{-t^2}dt; $$ it represented deviation from modularity of some other function. However I noticed ...
4
votes
1answer
283 views

Analytic continuation of a multiple contour integral

Let $W(t_1,\dotsc,t_n)$ a holomorphic function on some connected open set $U$ of $\mathbb C^n$. Let $\mathbf t^{(0)}$ a point of $U$. Assume that there exists a cycle $\gamma$ in $\mathbb C^m$ and a ...
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2answers
174 views

Analytic continuation for PI(1+z^(4^n))

How to do analytic continuation for following function? $$f(z) = \prod_{n=0}^{+\infty} {(1+z^{4^n})}$$ Evidently it satisfies $f(z)f(z^2)=\dfrac{1}{1-z}$...
5
votes
1answer
145 views

Ubiquity/scarcity of non-analytically continuable functions

Suppose f(z) is a power series with positive integer coefficients centered at zero and positive radius of convergence. What is the likelihood that f has a dense set of singularities on its circle of ...
0
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0answers
215 views

Analytical continuation of the reciprocal of the Zeta function

Is the reciprocal of the Zeta function analytically continuable? As $1/\zeta(n) = \sum_{n=1}^\infty \mu(n)/{n^s}$, this does not look obvious.
0
votes
1answer
221 views

Non-analyticity of convolution

I have posted a similar question in the past but let me make a final try in a simpler framework. Let $g \in C_0 ^\infty (\mathbb{R})$ be smooth and compactly supported. Define $$ f(x) = \int \big ...
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1answer
381 views

Functional equation of the alternating zeta function

Can one let me know about the functional equation of the alternating zeta function similar to the well known for the rieman function.
1
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1answer
443 views

continuation of the “n-th derivative” function [closed]

let $D_{\mathbb N}$ be the standard "n-th derivative" function is it possible to make a continuation of $D_{\mathbb N}$ to non integer values? i mean a function $D_{\mathbb R}$ such that $D_{\mathbb ...
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2answers
936 views

Does a bounded real function have an analytic continuation [closed]

Consider the function $f:[0,1]\rightarrow\mathbb{R}$, where $f$ is real-analytic on the open interval $(0,1)$ $f$ is bounded on the closed interval $[0,1]$ (ie. there is some constant $C$ such that ...
3
votes
1answer
283 views

Uniqueness of analytic continuation on a domain of C^n.

Hi. I have been struggling with this question for a while now. Given a domain $\Omega \subset \mathbb{C}^n$, $\Omega^\prime = \Omega \cap \mathbb{R}^n$, and an analytic function $f: \Omega ...
0
votes
1answer
419 views

Analytical continuation of a Dirichlet series with periodic coefficients

Fix a complex number s and a real number x, does there exist an analytic continuation of the Dirichlet series $L(s,x):=\sum_{k=1}^{\infty}\frac{\sin^2(2\pi k x)}{k^s}$ to the whole complex plane ...
6
votes
2answers
698 views

analytic continuations

Analytic/meromorphic continuation is a difficult problem in general. For "motivic L-functions", the idea of proving their analytic continuations by first proving their modularity goes back, I guess, ...