The analytic-continuation tag has no usage guidance.

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### Path lifting property of holomorphic unbranched map

Suppose $X$ is a Riemann surface and $ a\in\ X $ suppose $ \phi\in\mathcal O_a $ is a holomorphic function germ at $a.$ According to the theorem 7.8 of Forster's book Lectures on Riemann surfaces on ...

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**1**answer

164 views

### Analytic Continuation of Zeta-like function

Reading a paper about eta invariants I came across a zeta-like function.
I'm looking for the analytic continuation of $$\sum_{k=1}^\infty k(k+a)^{-s}$$ at $s=0$, where $a$ is positive.
In the paper ...

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30 views

### Relation between infinite product and regularized product

For a positive sequence $0\le\lambda_{1}\le\lambda_{2}\le\cdots$, consider an infinite product
\begin{equation*}
\prod_{i=1}^{\infty}\lambda_{i}:=\lim_{n\rightarrow\infty}\prod_{i=1}^{n}\lambda_{i}...

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291 views

### New/useful method for summation of divergent series?

Questions
$$ S(n,x) = x+e^x + e^{e^x} + e^{e^{e^x}} + \dots \text{$n$ times}$$
Also obeys (see background for argument):
$$ \frac{1}{2 \pi i} \oint e^{S(k,x)} \frac{\partial \ln(\frac{\int_0^\...

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**1**answer

116 views

### Analytic continuation of Euler product $\prod_{p} (1 - e^{-2 \pi i p \alpha}p^{-s})^{-1}$

Is anything useful known about the function defined by
\[
f(s, \alpha) = \prod_{p} (1 - e^{-2 \pi i p \alpha}p^{-s})^{-1} \quad ?
\]
Here, $\alpha$ is real. When $\alpha = 1$, this is certainly the ...

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votes

**1**answer

133 views

### Product representations of the Riemann zeta function

The Riemann zeta function, initially defined as
$$\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s} $$
for $\Re(s)>1$ also has infinite sum representations on larger domains. One such sum is
$$\zeta(s)=\...

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121 views

### Boundary behavior of power series vs. boundedness of partial sums

Let $f\left(z\right)=\sum_{n=0}^{\infty}a_{n}z^{n}$ be a power series with $0$s and $1$s as its coefficients ($a_{n}\in\left\{0,1\right\}$ for all $n$) with a radius of convergence of $1$. I call such ...

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100 views

### Analytic continuation of $\alpha(s)=\sum_{n=a+1}^\infty\frac{1}{(n^2-a^2)^s}$. Possibly related to Riemann Zeta function $\zeta(s)$?

I'm trying to find the analytic continuation for
$\alpha(s)=\sum_{n=a+1}^\infty\frac{1}{(n^2-a^2)^s} ,$
with $a\in \mathbb{N^+}$ and $s<1$. I need most likely only the values for $s=\frac{1}{2}-m$...

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**1**answer

107 views

### Analytic extension of the Hurwitz ζ function

For the purpose of formalisation in a theorem prover, I am looking for a simple definition of the analytic extension of the Hurwitz ζ function $\zeta(s,q)$ valid for all $s\in\mathbb{C}\setminus\{1\}$ ...

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**1**answer

280 views

### Analytic continuation of definite integral?

This question comes from physics...
We have the following functional (or function dose not matter):
$$S\left[x\right]:=\intop_{-\frac{t_{0}}{2}}^{+\frac{t_{0}}{2}}dt\,\mathcal{L}\left[x\right]\::\:{\...

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**1**answer

132 views

### Reconstructing analytic tetration with a complex height from a thinner set of points

This is a follow-up to my previous question An explicit series representation for the analytic tetration with complex height.
Recall the definition $(11)$ from there:
$$t(z) = \sum_{n=0}^\infty \sum_{...

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1k views

### An explicit series representation for the analytic tetration with complex height

Tetration is the next hyperoperation after more familiar addition, multiplication and exponentiation. It can be seen as a repeated exponentiation, similar to how exponentiation can be seen as a ...

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281 views

### Does an analytic continuation for a particular Leibniz series exist?

Define a Leibniz series as follows,
\begin{eqnarray*}
L(x) & = & \sum_{k=1}^{\infty}(-1)^{k}e^{-kx}\ln k,\ \ x>0
\end{eqnarray*}
I have two questions: (I) Is there an explicit formula for $...

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**1**answer

497 views

### Can an analytic function defined on a maximal torus be extended analytically to all the Lie group?

Let $G$ be a compact group and $T$ a maximal torus on $G$. Suppose $f$ is an analytic function defined on $T$. Is there an analytic function $F$ on $G$ whose restriction agrees with $f$ on $T$?

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226 views

### Analytic continuation of “composite” zeta function

Let us define the Dirichlet series $$\mathcal C(s):=\sum_{n\text{ composite}}\frac{1}{n^s},\quad P(s):=\sum_{p\text{ prime}}\frac{1}{p^s}.$$
They are absolutely convergent in the half-plane $\sigma>...

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105 views

### How many real-analytic forms exist on a real-analytic manifold?

I hope my question isn't too vague:
Let $M$ be a real-analytic compact manifold (say surface for simplicity). How big is $\Omega^k(M)$ (space of global real-analytic $k$-forms)?
Let me explain ...

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25 views

### Numerical test for unique continuation principle

Is there any numerical algorithm to test the unique continuation principle for the elliptic or parabolic equations?
Thank you very much.

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427 views

### Grothendieck topologies on $\mathbb{C}$

I would like to consider three sheaves $\mathcal{C}^0$, $\mathcal{H}$ and $\mathcal{S}$ on $\mathbb{C}$ (endowed with the euclidean topology): the first is the sheaf of continuous $\mathbb{C}$-valued ...

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439 views

### 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. I would like to ...

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### 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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49 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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100 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 $y^{-1}=(y_1^{-1},y_2^{-...

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**1**answer

491 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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249 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 $0<\...

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793 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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554 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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**1**answer

434 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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198 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

169 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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531 views

### Analytical continuation of the reciprocal of the Zeta function [closed]

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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**1**answer

260 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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**1**answer

398 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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**1**answer

481 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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**2**answers

1k 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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votes

**1**answer

360 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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**1**answer

480 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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### Analytic continuation of holomorphic functions

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, ...