The analytic-continuation tag has no usage guidance.

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### Analytic continuation of the double sum $\sum_{n,m\ge0}x^ny^mt^{nm}$

Define function $f(x,y,t)$ as the analytic continuation of the series
$$f(x,y,t)=\sum_{n,m\ge0}x^ny^mt^{nm}$$
This series definitely converges when all the arguments are small enough. My goal is to ...

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58 views

### Existence of analytic continuation of $f(z)=\sum{n^{\alpha}} z^n$ for fractional $\alpha$

The power series
$f(z)=\sum_{n \ge1}{n^{\alpha}} \cdot z^n$
has radius of convergence 1. For $\alpha \in \mathbb{N}$ it is easy to see that $f$ permits an analytic continuation to $\mathbb{C} ...

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301 views

### Analytic continuation for $L$-functions of elliptic curves

Let $E$ be an elliptic curve over a number field.
When $E$ has no CM and is a $\mathbb Q$-curve (i.e. it is $\overline{\mathbb Q}$-isogenous to all of its conjugates), it is nowadays known that $E$ ...

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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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65 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 ...

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378 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 ...

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217 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 ...

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505 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 ...

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508 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 ...

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316 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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176 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 $f(z)f(z^2)=\dfrac{1}{1-z}$...

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**1**answer

150 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 ...

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232 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.

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224 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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384 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.

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454 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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994 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 ...

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290 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 ...

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422 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 ...

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749 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, ...