Questions tagged [analytic-continuation]
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94
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What is known about the analytic continuation of Maz'ya's modified harmonic zeta function $\sum_{n=1}^{\infty} e^{-zH_n}$?
Question:
If we let $H_n = \sum_{k=1}^{n} \frac{1}{k}$ be the harmonic numbers then we can consider the modified zeta function
$$ f(z) = \sum_{n=1}^{\infty} e^{-zH_n } = \sum_{n=1}^{\infty} e^{-z(\ln(...
1
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0
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132
views
Analyticity of a function in two complex variables
Let $f$ be a function defined on $\mathbb{C}^2$ given by
$$
f(s,t)=\int\limits_{-\infty}^{\infty}dk_1 \int\limits_{-\infty}^{\infty}dk_2 \int\limits_{-\infty}^{\infty}dk_3 \frac{1}{\left(\sqrt{s}-k_1\...
7
votes
1
answer
508
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Help finding an analytic continuation
I am looking for the analytic continuation of
\begin{align*}
& f_m(v,w) := \sum\limits_{k,l=0}^\infty v^k w^l {k+l+m \choose k} {k+l+m \choose l} \ ,
\end{align*}
where $m \in \{1,2,...\}$ is ...
1
vote
0
answers
42
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analytic continuation of Krassner's analytic element on punctured disk
Let
$$
E=\{x\in\mathbb Q_p\mid |x|_p\le p^{-2}\}\setminus\left\{p^n\mid n\in\mathbb N,n\ge2\right\}
$$ and $g$ be an analytic element (in Krassner's sense) of
$$
F=\{x\in\mathbb Q_p\mid |x|_p<1\}\...
5
votes
0
answers
164
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Numerical analytic continuation/asymptotics
I posted this question, quite a while ago, on math.stackexchange.com, here. I received an interesting answer but not sufficiently accurate for my purposes, so I'm trying here.
I have a class of ...
4
votes
1
answer
466
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$\frac {f (0)}{2}+ \sum_{k=1}^{\infty}f (k)=\sum_{n=-\infty}^{\infty} \mathcal{L} \{ f \} (2 \pi i n)$
I obtained the very strange formula above and at begining I was just wanted know how to interpretate it. But now when I know what is this with help of @Carlo Beenakker, I am leaving it as a proof. BTW ...
2
votes
1
answer
260
views
Analytic continuation of a function on the half line
Consider the analytic function $f : (0,\infty) \to (0,\infty)$ given by
$$
f(x) = \bigg( \sum_{i=1}^n a_i b_i^{1/x} \bigg)^x
$$
where $n\in\mathbb N$, $a_i>0$ and $0<b_1<\ldots<b_n<1$. ...
5
votes
2
answers
293
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Functional equation of bounded analytic functions
Let $\mathbb{D}:=\{z \in \mathbb{C}|~|z|<1\}$. Consider $f,~g \in H^{\infty}(\mathbb{D}):=\{f\in\text{ Hol}(\mathbb{D}): \|f\|_{\infty}:= \sup_{z \in \mathbb{D}} |f(z)| < \infty\}$ such that $f^...
2
votes
1
answer
149
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Basic results concerning the intertwining operator in the $\mathrm{SL}_2$ case
I am reading [Ikeda, Tamotsu, On the location of poles of the triple L-functions]. On page 194, the author recalled some known results concerning $\operatorname{SL}_2$. I would like to know any ...
2
votes
0
answers
143
views
Meaning of the meromorphic continuation of intertwining operators
I am trying to make sure the meaning of the meromorphic continuation of the intertwining operators.
Assume we deal with a non-Archimedean field $F$ and just consider $G= SL_2$, for simplicity. We fix ...
4
votes
1
answer
233
views
Does the analytical continuation of $\sum f(n) x^n $ always have a branch cut if $f(z)$ has a pole?
I suspect the answer to the title question is 'no', but I'm hoping to find an explicit counterexample. Also, I am requiring that $\sum f(n) x^n $ has a finite radius of convergence, otherwise, the ...
8
votes
0
answers
241
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Switching the order of a summation and replacing a series by its analytical continuation
Background
A useful trick when trying to analyze a series $\sum_{n=0}^\infty f(n)$ is to expand $f(n)$ as some kind of series, swap the order of summation, and then evaluate the inner infinite sum. ...
0
votes
0
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131
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Gegenbauer polynomial relation with complex argument
Gegenbauer polynomials, $C_j^{\nu}(t)$, are defined to be the coefficient of $h^j$ in the expansion $(1-2ht +h^2)^{-\nu}$. It can be shown using [Higher Transcendental Functions, Vol 1, Harry Bateman, ...
0
votes
1
answer
54
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Analytic continuation of spline wavelets (reference request)
I would like to extend (cubic or higher degrees) spline wavelets to complex domain. First, does this continuation exist? Second, I appreciate it if anyone could point me to some references.
0
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0
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On "canonical" extensions of functions from integers to reals
Although this is essentially a port of my MathSE question, I think the users there tend to not understand how to interpret the questions from a higher perspective (and often too literally). This is ...
7
votes
0
answers
297
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Gottfried Helms' tetra-eta series
Here Gottfried Helms introduces the following fascinating divergent series
$$ T_2(x)=- \sum_{n=1}^\infty (-1)^n n^{n^x}$$
The terms don't go to zero, so technically the series does not converge ...
19
votes
6
answers
1k
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What is the relationship between $\sum_{n=0}^\infty f(n) x^n$ and $-\sum_{n=1}^\infty f(-n) x^{-n}$?
Background
Taking a relatively arbitrary combination of exponential and polynomial terms, for instance
$$\sum_{n=0}^\infty \left(n^{2}\sin\left(n\right)+n\cos\left(3n-2\right)\right)\cos\left(5n+1\...
4
votes
1
answer
142
views
Quantitative analytic continuation estimate for functions small except on a small set
This question arises as a variation of this question, which was helpfully answered in the negative. It turns out that for my application, a substantially weaker conjecture suffices, which fails to be ...
3
votes
3
answers
413
views
Quantitative analytic continuation estimate for a function small on a set of positive measure
The following conjecture about analytic functions arose as a way to show the asymptotic growth for certain PDE solutions. As I am unfamiliar with any results of this type, I thought I'd ask here.
In ...
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Analytic continuation in the Hardy class of analytic functions arcoss the boundary
I have the following question, which I need help addressing.
If $U\subset \mathbb{C}$ and $D\subset\mathbb{C}$, with $\partial U\cap\partial D=\Gamma\neq\emptyset$ of positive measure, are open ...
4
votes
1
answer
378
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How to correctly renormalize this function at the pole $x=1$? Evaluating: $\sum_{n=1}^{\infty} e^{e^n}$
So I was considering the divergent everywhere but 0 power series
$$ f(x) = \sum_{n=0}^{\infty} e^{e^n} x^n $$
Now one can do the following "questionable" manipulation
$$ f(x) = \sum_{n=0}^{\...
4
votes
0
answers
168
views
Mellin transform of the Bessel function $Y_n$ of order $n \geq 2$
The Mellin transform of the function $h$, locally integrable on $(0,\infty)$, is defined by
$$M[h,z] = \int_0^\infty t^{z-1} h(t) dt \tag{1}$$
For some functions $h$ the above integral is not ...
4
votes
0
answers
142
views
When is maximal analytic continuation a Zariski open set?
I have following question in mind. I apologize if this question is too obvious or naive.
Let $U$ be an Euclidian open set of $\mathbb{C}^*$ and $f:U\to U \times F\subset\mathbb{C}^4$ be an analytic ...
4
votes
1
answer
254
views
Derivative of Cauchy PV is equivalent to Hadamard regularization?
Let $\mathcal C$ and $\mathcal H$ denote the Cauchy principal value and Hadamard finite part. According to the Wiki:
$$
{\frac {\mathrm d}{\mathrm dx}}\left({\mathcal {C}}\int _{{a}}^{{b}}{\frac {...
3
votes
1
answer
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The monodromy in the proof of Little Picard via Klein's $J$
First of all, my question is in the context of formalising proofs in the proof assistant Isabelle/HOL, so my questions are a bit guided by what material we have in the complex analysis library there.
...
10
votes
1
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319
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On a variant of Carlson’s theorem
My question is on whether or not there exists some monotone strictly decreasing sequence of positive numbers $c_1>c_2>\ldots$ such that given any $f$ which is a uniformly bounded holomorphic ...
1
vote
0
answers
224
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Maximal analytic continuation
I'm currently reading about the concept of a “maximal analytic continuation” from Forster's book Lectures on Riemann Surfaces (see Section 7). There are a bunch of definitions to unpack before I can ...
5
votes
1
answer
247
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Are lacunary functions still lacunary over rings larger than $\mathbb{C}$?
Take a lacunary function of your choice ex:
$$ f(z) = z + z^2 + z^4 + \cdots = \sum_{k=0}^\infty z^{2^n} $$
Obviously this cannot really be analytically or meromorphically continued outside the unit ...
4
votes
1
answer
231
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Finding closed forms/related constants to a limit involving tetration
I was working on finding a series expression for a function $f: \mathbb{C} \rightarrow \mathbb{C}$ such that $f(x^y) = f(x)^{f(y)}$ along the way for construction of such a function I came across a ...
11
votes
1
answer
554
views
An extension of the Carlson's theorem in complex analysis
For the statement of Carlson's theorem please see,
https://en.wikipedia.org/wiki/Carlson%27s_theorem.
There is an extension of Carlson's theorem that says that the condition that $f$ needs to vanish ...
12
votes
1
answer
973
views
Divergent series summation beyond natural boundaries
I'm hoping to investigate the effects of divergent summation methods on series which cannot be analytically continued due to a dense set of singularities. At least a priori, it doesn't seem that a ...
4
votes
1
answer
283
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Is there a decision procedure for analytic continuation?
Let an analytic element be a power series associated with an open disc of the complex plane over which the series is convergent. W.l.o.g. assume the series is a Taylor expansion about the center of ...
3
votes
2
answers
259
views
Does this method analytically continue gap series series?
I was looking for ways to continue gap series, and it seemed to be that they could be continued outside of the boundary by simply turning
$$f(x)= \sum_{n=0}^\infty x^{n^k}$$
into
$$g(x) =- \sum_{n=1}^\...
4
votes
2
answers
173
views
Ordinary generating functions with finitely many singularities at algebraic numbers are rational
I have a proof of the following fact related to ordinary generating functions, and I was curious if it was known, as it seems plausible it is classically known:
"Let $\lambda_1,\ldots, \lambda_k$ ...
6
votes
0
answers
2k
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Do smooth cutoff functions analytically continue functions?
My goal is to prove (or disprove) that sufficiently smooth and quickly decaying cutoff functions being tacked on to a Taylor series correctly extend the radius of convergence to the analytic ...
11
votes
2
answers
875
views
Have any proposals been advanced for the analytic continuation of the divisor function?
While I was working on the evaluation of a certain series, the following limit came up:
\begin{align} \lim_{n \to 1} \frac{d(n)-1}{n(n-1)} &= \lim_{n \to 1} \frac{d'(n)}{2n-1} \\
&= d'(1) .\...
-1
votes
1
answer
268
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A question on assigning finite values to divergent sums involving expression of primes
We know the following:
$$\gamma=\lim_{n\to\infty }\left(\sum_{k=1}^n\frac{1}{k}-\ln(n)\right).$$
This could be a good candidate for renormalized sum of $\left(\sum_{k=1}^{\infty}\frac{1}{k}\right)$.
...
0
votes
1
answer
268
views
Analytic continuation of a periodic function on the real line
In the study of superconformal indices for certain quantum field theories, one encounters the elliptic $\Gamma$ function, which can be expressed as:
$$ \log \Gamma(z;\tau,\sigma)=\sum^{\infty}_{l=1}\...
2
votes
1
answer
530
views
On infinite sum containing logarithmic derivative of Zeta function and Möbius function:
Consider the following function:
$$F(s)= \sum_m \mu(m) \sum_n \frac{e^{-n/2}\zeta^\prime (mns)}{n \zeta(mns)}$$
Now, we can see, that function has simple poles ${\left[\frac{1}{n}\right]}_{n=1}^\...
1
vote
1
answer
286
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Continuing an analytic continuation of the Dirichlet $\eta$-function?
The Dirichlet $\eta$-function is defined as:
$$\eta(s) = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^s} \qquad \Re(s) > 0$$
and has the full analytical continuation:
$$\eta(s) = \sum_{n=1}^N \frac{(-1)^{...
5
votes
1
answer
642
views
Maximal analytic continuation of $\varphi(s)=\sum_{n\ge1} e^{-n^s}$
About 6 months ago I asked for an analytic continuation of $\varphi(s)=\sum_{n\ge1} e^{-n^s}.$
What's the maximal analytic continuation of $\varphi(s)?$
Doing this will help me better understand how ...
3
votes
0
answers
75
views
Continuation of $\sum \sigma_\nu(n) a(n) n^{-s}$ for $a(\cdot)$ coming from a half-integral weight form
In some of my work, I've run into a wall trying to understand whether a Dirichlet series has a meromorphic continuation or not. Let $f(z) = \sum_{n \geq 1} a(n) e(nz)$ be a half-integral weight ...
16
votes
1
answer
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What are some of the earliest examples of analytic continuation?
I'm wondering how Riemann knew that $\zeta(z)$ could be extended to a larger domain. In particular, who was the first person to explicitly extend the domain of a complex valued function and what was ...
1
vote
2
answers
356
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Global theory of holomorphic functions [closed]
I am trying to develop a theory explaining analytic continuation of a holomorphic function $f(z)$ defined on an open set $D \subset \mathbb{C}$. Recently, I was looking at the last chapter of Lars ...
2
votes
1
answer
270
views
Analytic continuation of convergent integral
I was trying to solve the following integral:
$$I = \oint _{|z|=1}\frac{dz}{2 \pi i z}\int_{0}^{\infty} dr \dfrac{e^{-\tfrac{r^2}{z^2}}r^{2n+1}}{z^2(z-1)} $$
The singular structure in the $z$ ...
4
votes
0
answers
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Tauberian theorem converse (Wiener-Ikehara)
Jacob Korevaar provides a nice converse to to Wiener-Ikehara tauberian theorem on p. 125 of his Tauberian Theory book:
For the non-decreasing, locally of bounded variation function $s$, if we have $\...
11
votes
1
answer
396
views
A density question for the Hilbert transform
Let $\mathscr Hf$ denote the Hilbert transform of a function $f$ defined on the real-line $\mathbb R$. Are the set of functions
$$ \{(f+\mathscr Hf)_{|_{(0,1)}}\,:\, f \in C^{\infty}(\mathbb R)\quad \...
2
votes
0
answers
129
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Question regarding proof of Mertens' estimates in Montgomery-Vaughan's "Multiplicative number theory"
I have been trying to read Theorem 2.7 of Montgomery-Vaughan's "Multiplicative number theory" volume 1, and there is an issue I have run into: in the proof of subpart (e), when they write
$$\...
1
vote
1
answer
158
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Are there variations of Ramaswami's formula for the analytic continuation of the Riemann zeta function?
On p. 286 of Borwein's paper entitled "Computational Strategies for the Riemann zeta function", the author mentions a formula due to Ramaswami: $$(1-2^{1-s})\zeta(s) = \sum_{n=1}^{\infty} \...
14
votes
1
answer
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What is the analytic continuation of $\varphi(s)=\sum_{n \ge 1} e^{-n^s}?$
My research has lead me to the following function that I'm trying to continue. 3 Months ago I posted this question to MSE, and have placed 3 bounties on the question, but haven't received an answer, ...