Questions tagged [analytic-continuation]
For questions related to analytic continuation
541
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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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15
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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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13
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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\...
- 6,613
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2
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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 ...
- 3,907
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10
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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 ...
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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{...
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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 ...
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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 ...
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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(...
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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}$$
Where $a_n$ is a sequence of distinct positive integers,
such that
$f(s)$ is analytic for all $Re(s) > 1$
$f(s)$ has a simple ...
- 14.6k
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Can we substitute values after analytic continuation
Motivating example:
Consider the following $f(z)=\sum_{n\geq 0}A_n(z)$, and $g(z)=\sum_{n\geq 0} B_n(z)$.
Lets say $A_n(z)=\frac{(-1)^n}{(2n+1)^z}$ and $ B_n(z)=\frac{(-1)^{n}}{(n+1)^z}$
Then $f(z)=\...
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Is this a valid analytic continuation of ζ(s) for 0 < Re(s) < 1?
Given $\zeta(s) = 1^{-s} + 2^{-s} + 3^{-s}...$
$2^{-s}\zeta(s) = 2^{-s} + 4^{-s} + 6^{-s}...$
$\zeta(s)(1-2^{1-s})=1^{-s} - 2^{-s} + 3^{-s} - 4^{-s}...$ which converges for $0 < \Re(s) < 1$.
Is ...
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Solutions for $Re(z) > 2$ ,$f(z) = \sum_{1<i}^{\infty} f(i)^{\dfrac{z-i+1}{i}}$?
Consider the functions defined by the equations and conditions
$$Re(z) > 2$$
$$f(i+1) > f(i)-2$$
$$ f(z) = \sum_{1<i}^{\infty} f(i)^{\dfrac{z-i+1}{i}}$$
Does that make sense ? Does that ...
- 14.6k
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Convergence in a divergent double sum by substituting analytic continuation values
There is a definition of the Gram series of R(x) where R(x) is a term in the exact formula of the prime counting function defined as
$$R(x) = 1 + \sum_{k=1}^{\infty}\frac{(log(x))^k}{k!k\zeta(1+k)} $$
...
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How can the domains of all variables of this hyperoperation function be extended to the entire complex plane?
When generalizing addition, multiplication, and exponentiation, there exists a certain hierarchy upon which these operators can be placed: the hyperoperator hierarchy. Starting with succession, each ...
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Is there some general technique to turn the operation of function into a continuous transformation from input to output?
To be precise:
Given a function $f(x)$ I want a function $g(t, x)$, $x,t \in \mathbb{R}$ or $\mathbb{C}$, such that $g(t, x)$ equals the $t$-time application of $f$ to the value $x$ for $t \in \mathbb{...
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Analytical continuation of $-\frac{t^w}{\ln(2)}\sum_{k=-\infty}^\infty \frac{t^{w k}}{e^{\pi i w(1+2k)}-1}$
I'm interested in obtaining the analytical continuation of the following function. Let $w = \frac{2 \pi i}{\ln(2)}$, then let
$$F(t) = -\frac{t^{w/2}}{\ln(2)}\sum_{k=-\infty}^\infty \frac{t^{w k}}{e^{\...
- 1,043
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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)=\...
- 840
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2
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106
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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, ...
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96
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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}$ ...
- 475
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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....
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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(...
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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 ...
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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.
...
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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)}$$
The ...
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Roots and analytic continuation of $\zeta(s)=\sum_{n>0} \frac{H_n^{-s}}{n} $?
Let
$$\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$$
??
...
- 14.6k
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2
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Analytic continuation of a lambert series $f(z)=\sum_{n=1}^\infty \frac{2^n z^n}{1-(z/2)^n}$?
Let's define a function for complex $z$ :
$$f(z)=\sum_{n=1}^\infty \frac{2^n z^n}{1-(z/2)^n}$$
Lambert series often have an analytic continuation and can even be entire functions so the poles can be ...
- 14.6k
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Polynomial Result extended to complex argument
So, it is known that the Gegenbauer polynomial of degree $\nu$ and order $j$, $C_{j}^{\nu}(x)$, are such that they satisfy the following property
\begin{equation}
| C_{j}^{\nu}(x) | \leq C_{j}^{\nu}(1)...
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1
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Closed form of series $\sum_{n=\frac {1}{2}}^{\infty}\Gamma (1-s,xn)e^{xn}n^{s-1}$
I'm trying to get some closed form solution of $\sum_{n=\frac {1}{2}}^{\infty}\Gamma (1-s,xn)e^{xn}n^{s-1}$. To be precise gotten closed form should be analytic continuation for any given $x$. There ...
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Analytic continuation of $I(s)=\frac{1}{\Gamma(s)}\int^\infty_0 f(x)x^{s-1}\, dx$
This is Exercise 17 of Chapter 6 from Stein and Shakarchi Complex analysis
Let $f$ be an infinitely differentiable function on $\mathbb{R}$
that has compact support, or more generally, let $f$ ...
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Analytic continuation of mandelbrot function
Is that will work as analytic continuation for mandelbrot function, if not why?
Mandelbrot function can be defined with Catalan numbers as $\sum_{n=1}^{\infty}\frac {(2n-2)!}{(n-1)!(n)!}x^n$
From (...
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Analytic continuation of Bessel series
What is the analytic continuation of $$f(x)=\sum_{n=1}^\infty\frac{2K_1(2\sqrt{n^x})}{\sqrt{n^x}}$$
where $K_1$ is a modified Bessel function of the second kind.
This converges for real $x>0.$
I ...
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Why does this give the correct analytic continuation?
Consider some kernel function $f(n,s)$ with $\sum_n^\infty f(n,s)=f(s).$ Say an extension of $f$ is desired via analytic continuation. I'm curious about examples when that analytic continuationof $f(s)...
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Meaning of $\zeta(1-s)$ in Riemann Zeta function?
I've been looking at one of the Analytic Continuations of the Zeta function, the Riemann Zeta function:
$$\zeta(s) = 2^s \pi^{s-1} \sin \left(\dfrac{\pi s}2\right) \Gamma(1-s) \zeta(1-s)$$
I ...
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Show two analytic functions are related by a constant
The problem I'm working on says this: Suppose $f, g$ are two analytic functions on an open set containing $\overline{D}$, where $D = \{ z \in \mathbb{C} : |z| < 1 \}$ is the open disc in $\mathbb{C}...
- 8,599
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two-times analytic continuation of the domain
Lets imagine that D - given domain (strip) on upper half plane H, and G - its image (connected) under mapping f with required properties. The image has curved side (not stright line, like, polygons!). ...
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How is the substitution $t \mapsto it$ an analytic continuation?
In quantum theory it is common to use what is known as Wick rotation, which says for a function defined over $\mathbb{R}^n$, for at least one of the variables, make the substitution $t \mapsto it.$ So ...
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Convergence of definite integral and its analytic continuation
I have some very serious question (which truly isn't a joke) about convergence of definite integral in the form of $\int_{x}^{\infty}f (t)dt$, and its analytic continuation. To start, let's say that $...
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conformal mapping from unit disk to equilateral triangle
Let $$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$$ be a conformal map from the unit disc $B(0,1)$ to the equilateral ...
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Analytic continuation of $f(z) = \sum_{n=1}^{\infty} \frac{1}{2^n (z-e^{2\pi i r_n})}$
Let $\{r_k:k\in\mathbb{N}\}$ be a counting set of rational numbers in $[0,1]$. Show that $$f(z) = \sum_{k=1}^{\infty} \frac{1}{2^k (z-e^{2\pi i r_k})}$$ is holomorphic on $U_1=\{z\in \mathbb{C} : |z| &...
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Roots and analytic continuation of $T(s)=\sum_{n>0} (n^s + n^{-s})^{-1} $?
Let $s$ be a complex number.
$$T(s)=\sum_{n>0} (n^s + n^{-s})^{-1} $$
This is well defined for $Re(s) > 1$.
It seems $T(s) = T(-s)$ but then again we have it only defined for $Re(s)>1$ for ...
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Can the complex function $f(z) = \sqrt[n]{r^n - z^n}$ be analytically continued to fill the complex plane?
I have the complex function $f(z) = \sqrt[n]{r^n - z^n}$, where $z$ is a complex variable, $r$ is a positive real number, and $n$ is any non-zero real number. For any value of $n$ besides 1, this ...
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New $ \overset {s}{\underset{k=x}{ \lower 3pt { \LARGE\Xi}}} $ operator and Möbius function
At the beginning I will explain some concepts and at the end I will ask the specific question.
Let's consider some operator $ \overset {s}{\underset{k=x}{ \lower 3pt {
\LARGE\Xi}}} $ on function $f ...
- 388
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50
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Deducing that an analytic function $f$ in a simply connected domain $\Omega$ is the derivative of an analytic function $F:\Omega\to\mathbb{C}$
Let $\Omega$ be a simply connected domain, i.e. open and connected subset of $\mathbb{C}$ and $f$ be analytic in $\Omega$. I am trying to understand how you can deduce that $f(z) = \frac{d}{dz}F(z)$ ...
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The relationship between interior and exterior Riemann maps
Let $U\subset\widehat{\mathbb{C}}$ be a compact Jordan domain with $\partial U$ connected. Is there a relationship between the exterior Riemann map associated to $U$,
$$\varphi:\mathring{(\mathbb{D}^c)...
5
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1
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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 ...
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138
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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\...
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