Questions tagged [analytic-continuation]

For questions related to analytic continuation

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How do we prove this nested radical solution?

I’m submitting this question because my previous one was perhaps too verbose and did not get the point very quickly. A well known $\sqrt{2}$ nested radical has the following solutions as found in the ...
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A Nested Radical Arising from a Nonlinear Recurrence

I have been looking into analytic continuation of regular recurrence relations into the negative, real, and complex domains ($\mathbb{N} \mapsto \mathbb{Z}, \mathbb{R},\mathbb{C}$). In doing so, we’ve ...
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Prove that this limit is $-2$, which is a trivial zero of the Riemann zeta function.

The numerical evaluation of this formula works in Mathematica. Let: $$s=1$$ and truncate, at some large number, the limit involving the Harmonic numbers: $$f(x)= \lim_{m \rightarrow \infty}H_{m}^{x}$$ ...
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Analytic continuation of sums of Hypergeometric $_3F_2$ function pairs

While investigating the relationship $$\Gamma\left[f,s\right]\zeta(s)=\Gamma\left[\hat{f},1-s\right] \zeta(1-s)\tag{1}$$ where $\Gamma\left[f,s\right]$ denotes the Mellin transform $$\Gamma\left[f,s\...
Steven Clark's user avatar
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Analytic continuation for $\zeta(-x)$ vs $\frac{1}{(1-x)^2}$ at $x=1$

Everyone and their grandma has seen youtube videos about the analytic continuation of the series definition of the zeta function. My question is regarding another function whose series expansion seems ...
Gappy Hilmore's user avatar
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Dense subspace for arbitrary powers of a positive operator

Let $\mathcal{H}$ be a Hilbert space, and let $\Delta$ be a positive, self-adjoint, unbounded operator. I think I can prove the following statement, but I find it quite surprising, and I can't find it ...
Jon's user avatar
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Overlap in Implicit Function Theorem

Consider $f_1(t,x_1,\ldots,x_n),\ldots, f_n(t, x_1,\ldots, x_n)$ for complex analytic functions $f_i$ around $\vec{a}=(t', a_1, \ldots, a_n)$ and $\vec{b}=(t^*, b_1,\ldots, b_n)$, such that $f_i(\vec{...
leftIdeal's user avatar
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249 views

Conditions to calculate an integral through a series expansion

Let $f,g:\mathbb{R}\longrightarrow\mathbb{R_+}$ be Lebesgue integrable functions. We can show that if $g$ has compact support and $f$ has a Maclaurin series that converges absolutely in the support of ...
P.S. Dester's user avatar
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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 ...
Haidara's user avatar
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Lipschitz functions (Theorem 1.4 of Condenser Capacities and Symmetrization in Geometric Function Theory)

I am struggling to understand the second part of the following proof of Theorem 1.4 of the book "Condenser Capacities and Symmetrization in Geometric Function Theory" by Vladimir N. Dubinin (...
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Regularity of Cauchy-like integral

I'm trying to understand whether a certain integral representation is holomorphic. Specifically, let's assume that $f$ belongs to the Schwartz space $\mathcal S(\mathbb R)$, and consider a function $g(...
sword's user avatar
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Looking for a specific zeta function.

I am looking for a zeta function $$ f(s) = \sum \frac{1}{a_n^s}$$ Where $a_n$ is a sequence of distinct positive integers, such that $f(s)$ is analytic for all $Re(s) > 1$ $f(s)$ has a simple ...
mick's user avatar
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Can we substitute values after analytic continuation

Motivating example: Consider the following $f(z)=\sum_{n\geq 0}A_n(z)$, and $g(z)=\sum_{n\geq 0} B_n(z)$. Lets say $A_n(z)=\frac{(-1)^n}{(2n+1)^z}$ and $ B_n(z)=\frac{(-1)^{n}}{(n+1)^z}$ Then $f(z)=\...
Dqrksun's user avatar
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Is this a valid analytic continuation of ζ(s) for 0 < Re(s) < 1?

Given $\zeta(s) = 1^{-s} + 2^{-s} + 3^{-s}...$ $2^{-s}\zeta(s) = 2^{-s} + 4^{-s} + 6^{-s}...$ $\zeta(s)(1-2^{1-s})=1^{-s} - 2^{-s} + 3^{-s} - 4^{-s}...$ which converges for $0 < \Re(s) < 1$. Is ...
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Solutions for $Re(z) > 2$ ,$f(z) = \sum_{1<i}^{\infty} f(i)^{\dfrac{z-i+1}{i}}$?

Consider the functions defined by the equations and conditions $$Re(z) > 2$$ $$f(i+1) > f(i)-2$$ $$ f(z) = \sum_{1<i}^{\infty} f(i)^{\dfrac{z-i+1}{i}}$$ Does that make sense ? Does that ...
mick's user avatar
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Convergence in a divergent double sum by substituting analytic continuation values

There is a definition of the Gram series of R(x) where R(x) is a term in the exact formula of the prime counting function defined as $$R(x) = 1 + \sum_{k=1}^{\infty}\frac{(log(x))^k}{k!k\zeta(1+k)} $$ ...
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How can the domains of all variables of this hyperoperation function be extended to the entire complex plane?

When generalizing addition, multiplication, and exponentiation, there exists a certain hierarchy upon which these operators can be placed: the hyperoperator hierarchy. Starting with succession, each ...
Heisenberg2010's user avatar
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Is there some general technique to turn the operation of function into a continuous transformation from input to output?

To be precise: Given a function $f(x)$ I want a function $g(t, x)$, $x,t \in \mathbb{R}$ or $\mathbb{C}$, such that $g(t, x)$ equals the $t$-time application of $f$ to the value $x$ for $t \in \mathbb{...
apirogov's user avatar
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Analytical continuation of $-\frac{t^w}{\ln(2)}\sum_{k=-\infty}^\infty \frac{t^{w k}}{e^{\pi i w(1+2k)}-1}$

I'm interested in obtaining the analytical continuation of the following function. Let $w = \frac{2 \pi i}{\ln(2)}$, then let $$F(t) = -\frac{t^{w/2}}{\ln(2)}\sum_{k=-\infty}^\infty \frac{t^{w k}}{e^{\...
Caleb Briggs's user avatar
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Analytic continuation of a series $\Psi(s)$?

In the Digital Library of Mathematical Functions they list some analytic continuations of modified Bessel functions. I'm highly interested in the analytic continuation of this function: $$ \Psi(s)=\...
John Zimmerman's user avatar
3 votes
2 answers
106 views

Regularizing infinite sum over $\sqrt{n^2+a^2}$

I am aware that one can use zeta function regularization to obtain \begin{equation} \sum_{n\in \mathbb{N}}n = -\frac{1}{12} \end{equation} I am wondering if it is possible to regularize a similar sum, ...
Kaixiang Su's user avatar
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1 answer
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Existence of holomorphic $f$ with $(f(z))^4=z^4+4$

Let $\Omega=\mathbb{C}\setminus K$ where $K$ is a compact connected set containing $1+i$, $1-i$, $−1+i$, and $−1−i$. I want to prove that there exists a holomorphic function $f:\Omega\to\mathbb{C}$ ...
Anon's user avatar
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Rudin's proof of Monodromy theorem.

The following is Theorem 16.15 in Rudin's RCA. After some searching, I figured out that there are a lot of different forms of the Monodromy theorem. So I wanted to attach the statement and proof of it....
Jiya's user avatar
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Analytic continuation of integral representation as Cauchy principal value

Suppose I have the following formula, \begin{equation} \mathrm{P}\int_{-\infty}^{\infty} f(x,q) dx = F(q), \end{equation} for all $q\in\mathbb{R}$, where P stands for the Cauchy principal value. If $F(...
norio's user avatar
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what are the branch points and branches of $g(z)=(z+ \sqrt{z})^{1/3}$?

And what if we for example shifted one of the roots, eg $f(z)=(z+ \sqrt{z-3})^{1/3}$? I already asked a more extensive version of this question here Branch cut/ points for square roots inside cubic ...
Noam's user avatar
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Branch cut/ points for square roots inside cubic roots- incorrect branching by mathematica or my mistake?

There's a lot of great information here about understanding the branch cuts and branch points of functions of the form ( for example ) $(z^3+1)^{1/2}$, sums of simple roots and products thereof. ...
Noam's user avatar
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Help with a step on a proof of the maximum principle for harmonic functions over a connected region

I know how to prove that if $f:A \rightarrow \mathbb{C}$ is a harmonic function (here $A$ is an open connected region in $\mathbb{C}$) such that $|f|$ admits a local maximum on a point $z_0\in A$, ...
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Analytical continuation of $\sum_{n=-\infty}^\infty \frac{\csc\left(\pi z(n+\frac{1}{2})\right)}{\Gamma\left(1+ z(n+\frac{1}{2})\right)}$

I'm interested in finding the analytical continuation of the function $$F(z) = \sum_{n=-\infty}^\infty \frac{\csc\left(\pi z(n+\frac{1}{2})\right)}{\Gamma\!\left(1+ z(n+\frac{1}{2})\right)}$$ The ...
Caleb Briggs's user avatar
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Roots and analytic continuation of $\zeta(s)=\sum_{n>0} \frac{H_n^{-s}}{n} $?

Let $$\zeta(s)=\sum_{n>0} \frac{H_n^{-s}}{n} $$ Where $H_n$ are the harmonic numbers. This is well defined for $\Re (s)>1$. But what about analytic continuation? And where is $$\zeta(s) = 0$$ ?? ...
mick's user avatar
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Analytic continuation of a lambert series $f(z)=\sum_{n=1}^\infty \frac{2^n z^n}{1-(z/2)^n}$?

Let's define a function for complex $z$ : $$f(z)=\sum_{n=1}^\infty \frac{2^n z^n}{1-(z/2)^n}$$ Lambert series often have an analytic continuation and can even be entire functions so the poles can be ...
mick's user avatar
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Polynomial Result extended to complex argument

So, it is known that the Gegenbauer polynomial of degree $\nu$ and order $j$, $C_{j}^{\nu}(x)$, are such that they satisfy the following property \begin{equation} | C_{j}^{\nu}(x) | \leq C_{j}^{\nu}(1)...
Jason Curran's user avatar
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Closed form of series $\sum_{n=\frac {1}{2}}^{\infty}\Gamma (1-s,xn)e^{xn}n^{s-1}$

I'm trying to get some closed form solution of $\sum_{n=\frac {1}{2}}^{\infty}\Gamma (1-s,xn)e^{xn}n^{s-1}$. To be precise gotten closed form should be analytic continuation for any given $x$. There ...
Wreior's user avatar
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2 votes
1 answer
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Analytic continuation of $I(s)=\frac{1}{\Gamma(s)}\int^\infty_0 f(x)x^{s-1}\, dx$

This is Exercise 17 of Chapter 6 from Stein and Shakarchi Complex analysis Let $f$ be an infinitely differentiable function on $\mathbb{R}$ that has compact support, or more generally, let $f$ ...
Paradox's user avatar
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Analytic continuation of mandelbrot function

Is that will work as analytic continuation for mandelbrot function, if not why? Mandelbrot function can be defined with Catalan numbers as $\sum_{n=1}^{\infty}\frac {(2n-2)!}{(n-1)!(n)!}x^n$ From (...
Wreior's user avatar
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Analytic continuation of Bessel series

What is the analytic continuation of $$f(x)=\sum_{n=1}^\infty\frac{2K_1(2\sqrt{n^x})}{\sqrt{n^x}}$$ where $K_1$ is a modified Bessel function of the second kind. This converges for real $x>0.$ I ...
John Zimmerman's user avatar
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Why does this give the correct analytic continuation?

Consider some kernel function $f(n,s)$ with $\sum_n^\infty f(n,s)=f(s).$ Say an extension of $f$ is desired via analytic continuation. I'm curious about examples when that analytic continuationof $f(s)...
John Zimmerman's user avatar
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Meaning of $\zeta(1-s)$ in Riemann Zeta function?

I've been looking at one of the Analytic Continuations of the Zeta function, the Riemann Zeta function: $$\zeta(s) = 2^s \pi^{s-1} \sin \left(\dfrac{\pi s}2\right) \Gamma(1-s) \zeta(1-s)$$ I ...
Runsva's user avatar
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2 answers
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Show two analytic functions are related by a constant

The problem I'm working on says this: Suppose $f, g$ are two analytic functions on an open set containing $\overline{D}$, where $D = \{ z \in \mathbb{C} : |z| < 1 \}$ is the open disc in $\mathbb{C}...
AJY's user avatar
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two-times analytic continuation of the domain

Lets imagine that D - given domain (strip) on upper half plane H, and G - its image (connected) under mapping f with required properties. The image has curved side (not stright line, like, polygons!). ...
Nash's user avatar
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How is the substitution $t \mapsto it$ an analytic continuation?

In quantum theory it is common to use what is known as Wick rotation, which says for a function defined over $\mathbb{R}^n$, for at least one of the variables, make the substitution $t \mapsto it.$ So ...
CBBAM's user avatar
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Convergence of definite integral and its analytic continuation

I have some very serious question (which truly isn't a joke) about convergence of definite integral in the form of $\int_{x}^{\infty}f (t)dt$, and its analytic continuation. To start, let's say that $...
Wreior's user avatar
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conformal mapping from unit disk to equilateral triangle

Let $$f(z) = C \int_{0}^{z} \frac{1}{(1-\zeta^3)^{\frac{2}{3}}}d\zeta$$ with $$C \int_{0}^{1} \frac{1}{(1-x^3)^{\frac{2}{3}}}dx=1$$ be a conformal map from the unit disc $B(0,1)$ to the equilateral ...
syphracos's user avatar
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1 answer
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Analytic continuation of $f(z) = \sum_{n=1}^{\infty} \frac{1}{2^n (z-e^{2\pi i r_n})}$

Let $\{r_k:k\in\mathbb{N}\}$ be a counting set of rational numbers in $[0,1]$. Show that $$f(z) = \sum_{k=1}^{\infty} \frac{1}{2^k (z-e^{2\pi i r_k})}$$ is holomorphic on $U_1=\{z\in \mathbb{C} : |z| &...
Quotenbanane's user avatar
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6 votes
1 answer
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Roots and analytic continuation of $T(s)=\sum_{n>0} (n^s + n^{-s})^{-1} $?

Let $s$ be a complex number. $$T(s)=\sum_{n>0} (n^s + n^{-s})^{-1} $$ This is well defined for $Re(s) > 1$. It seems $T(s) = T(-s)$ but then again we have it only defined for $Re(s)>1$ for ...
mick's user avatar
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Can the complex function $f(z) = \sqrt[n]{r^n - z^n}$ be analytically continued to fill the complex plane?

I have the complex function $f(z) = \sqrt[n]{r^n - z^n}$, where $z$ is a complex variable, $r$ is a positive real number, and $n$ is any non-zero real number. For any value of $n$ besides 1, this ...
Lawton's user avatar
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New $ \overset {s}{\underset{k=x}{ \lower 3pt { \LARGE\Xi}}} $ operator and Möbius function

At the beginning I will explain some concepts and at the end I will ask the specific question. Let's consider some operator $ \overset {s}{\underset{k=x}{ \lower 3pt { \LARGE\Xi}}} $ on function $f ...
Wreior's user avatar
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Deducing that an analytic function $f$ in a simply connected domain $\Omega$ is the derivative of an analytic function $F:\Omega\to\mathbb{C}$

Let $\Omega$ be a simply connected domain, i.e. open and connected subset of $\mathbb{C}$ and $f$ be analytic in $\Omega$. I am trying to understand how you can deduce that $f(z) = \frac{d}{dz}F(z)$ ...
Cartesian Bear's user avatar
2 votes
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The relationship between interior and exterior Riemann maps

Let $U\subset\widehat{\mathbb{C}}$ be a compact Jordan domain with $\partial U$ connected. Is there a relationship between the exterior Riemann map associated to $U$, $$\varphi:\mathring{(\mathbb{D}^c)...
Andrew Graven's user avatar
5 votes
1 answer
208 views

Numerical analytic continuation for $\tanh$-like function

I have a class of functions $f(z) \in \mathcal{F}$ of which I know the Taylor series at $z=0$ up to some free to choose order $N$. I am trying to numerically estimate the value of $f(z)$ outside its ...
lcv's user avatar
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Computation on converging alternating zeta-like series by analytic continuation

I found that to compute an converging alternating zeta-like series, I can convert it into the linear combination of zeta functions in its analytic continuation domain, e.g., I can calculate $\sum_{n\...
Eddie Lin's user avatar
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