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Questions tagged [analytic-continuation]

For questions related to analytic continuation

6
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318 views

Why do the Jacobi theta functions have a natural boundary?

The Jacobi theta functions, like $$ \theta_3(z,q)=1+2\sum_{n=0}^\infty q^{n^2}\!\cos(2nz) , $$ look relatively innocent in how they handle the 'nome' $q$, a complex parameter that shapes the ...
6
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0answers
612 views

Show that $f(z)=\sum_{n=0}^{\infty}z^{2^n}$ can't be analytically continued past the unit disk.

I'm reading the problems of Stein and Shakarchi's Complex Analysis, Chapter 2 Problem 1 asks to show that $$f(z)=\sum_{n=0}^{\infty}z^{2^n}$$ cannot be analytically continued past the unit disk. (...
5
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125 views

Is $\int_0^\infty \frac{dt}{e^t-xt}$ analytic continuation of $\sum_{k=1}^\infty \frac{(k-1)!}{k^k} x^{k-1}$?

The following power series apparently converges only for $-e \leq x <e$: $$f(x)=\sum_{k=1}^\infty \frac{(k-1)!}{k^k} x^{k-1}$$ We can use it to define a real function $f(x)$, analytic in that ...
4
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0answers
34 views

Translation of integration contour

I have an argument where, with a meromorphic function $\phi$ satisfying $\phi(s) = \phi(1-s)$, the following equality appears: $$\int_{(2)} \phi(s) y^{-s} \, ds = \int_{(1/2)} \phi(s) y^{-s} \, ds$$ ...
4
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0answers
87 views

Is there a natural topology in which the analytic continuation of a series converges everywhere it is defined?

The expression $$ \sum_{n=1}^\infty n = -\frac{1}{12} $$ has been met with considerable controversy. Previous questions (such as this and this) searching for a way of rationalizing the formal ...
4
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0answers
142 views

Black magic behind Padé approximation

In condensed matter physics one usually calculates the so-called Matsubara Green's function in the set of discrete points $G (i \omega_n)$, where $$ \omega_n = (2n+1) \pi T $$ Physically significant ...
4
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44 views

Analyitc Continuation of Partial $\zeta$ function

Let $A\subset \mathbb{N}$. Consider the partial $\zeta$ function $$\sum_{a\in A} \frac{1}{a^s}$$ We know this converges for Re$(s)>1$. Under what conditions of $A$, can this be analyically ...
3
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25 views

Meromorphic continuation of the multifactorial

The double factorial $z!!$, like the normal factorial function, can be extended to the complex plane using $$ z!! = 2^{z/2} \left(\frac{\pi}{2}\right)^{(\cos\pi z-1)/4} \left(\frac{z}{2}\right)! $$ ...
3
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0answers
41 views

Dirac delta from poles of a function

Suppose we are given the simple expression $$ F(k) = \frac{1}{E^2-E(k)^2} $$ which has a pole when $E^2 = E(k)^2$ and where $E, E(k)$ are real numbers. When working with this expression (e.g. inside ...
3
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72 views

Riemann zeta meromorphic cont. using Abel summation formula

In Stein&Shakarchi, Complex Analysis, chapter 6, problem 2-3 (p. 180), they hint at a method to meromorphically continue the zeta function to the entire complex plane. I can see from Abel's ...
3
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120 views

What is the relationship between the asymptotic series outside the radius of convergence and the analytic continuation?

In his post about computing the zeta function by smooth cutoff regularization of the series $\sum\frac{1}{n^2}$, Terry Tao shows how $-{1\over 12}$ is the finite part of the asymptotic series for the ...
3
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0answers
143 views

Comparison of entire functions with $f=cg$

Let $f,g:\mathbb C\to\mathbb C$ be holomorphic with $|f(z)|\leq|g(z)|$ for all $z\in\mathbb C$. Show that there is a $c\in\mathbb C$ with $|c|\leq 1$ such that $f=cg$. If $g\equiv 0$ then we ...
2
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75 views

For what real part of $s$ as a function of $q$ is the Euler-Maclaurin formula a valid analytic continuation of the Riemann zeta function?

The familiar formula for the Riemann zeta function: $$\zeta(s)=\lim\limits_{k \rightarrow \infty} \left( \sum\limits_{n=1}^{n=k} \frac{1}{n^s}\right) \mbox{ is true for } \Re(s)>1$$ adding one ...
2
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44 views

Why should the series representation of the zeta function know about its analytic continuation?

In physics, when we calculate the renormalized sum of $S=\sum_{n=1}^\infty n$, we usually use an exponential regulator and instead first calculate $$S_\epsilon = \sum_{n=1}^\infty ne^{-\epsilon n} = ...
2
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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}...
2
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130 views

Is this evidence that $g(x)$ can be analytically continued?

Argument Let, $f(x) = \sum_{r=1}^\infty \mu(r) x^r$ where $\mu(r)$ is the mobius function. Hence, $$ f(x) + f(x^2) + \dots = x$$ Now, let $x \to x^2$: $$ 0+f(x^2) + 0+\dots = x^2$$ Similarly $x ...
2
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0answers
161 views

maximal analytic continuation of a holomorphic germ on riemann surface

This is related to Forster, Riemann surfaces Exercise 7.1 Let $X$ be a riemann surface and $a\in X$ be a point with $\phi\in O_{X,a}$ where $O_{X,a}$ is the set of holomorphic germs at $a$. Let $Y$ ...
2
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0answers
45 views

For a solution to an ODE, when do we apply analytic continuation at a point where ODE is undefined?

As a continuation of this question, the given answer states that at a point where the ODE is undefined, we apply "analytic continuation" (if this is the correct terminology) to the solution so that ...
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47 views

Computing the value of a spectral zeta function at zero

Suppose I have some operator $\mathcal{O}$ with eigenvalues $\lambda_n$ and the corresponding zeta function $$ \zeta_\mathcal{O}(s) = \sum_{n=1}^\infty \lambda_n^{-s} \,.$$ I want to compute the value ...
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51 views

Was in the literature the function $\sum_{n=1}^\infty\frac{1}{N_n^s}$, where $N_n$ is the $n$th primorial number?

I am curious to know if this function was in the literature, and mainly if it is possible define an analytic extension out of $\Re s>0$. Let $N_n$ the $n$th primorial number, and $s=\sigma+it$ ...
2
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0answers
93 views

Is there a generalization of the convergence radius of analytic functions to higher dimension?

Lets start with a simple example: We take the analytic function $$ f:R -> R , f(x)= \frac{1}{1+x^2}$$ if we do a Taylor expansion around 0 we get a power series with convergence radius of 1. How ...
2
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88 views

A question about continuation

My question is quite general, but let me explain it with a specific example. Suppose I have a PDE: $$\frac{d^2}{d\rho^2}f(\rho)+\frac{3}{\rho}\frac{d}{d\rho}f(\rho)+U[f(\rho)]=0,\tag{1}$$ where $\rho\...
2
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0answers
78 views

Dirichlet series associated to smallest prime factors

If we denote by $p_1(n)$ the smallest prime factor of $n$, has the Dirichlet series $$\sum_{n=2}^\infty\frac{p_1(n)}{n^s}$$already been studied? What is known about its analytic continuation?
2
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0answers
164 views

Given an analytic continuation along $\gamma$ such that $R(t)\equiv \infty$ for some $t$, then $R(s)\equiv \infty$ for each $s\in [0,1]$

Definition: A function element is a pair $(f,U)$ where $U$ is a region and $f$ is an anaytic function on $U$. For a given function element $(f,U)$ define the germ of $f$ at $a$ to be the ...
2
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0answers
91 views

How to analytically continue this function?

I was wondering if it would be possible to get an analytically continuation of the following function: $$ J(x) = \sum_{r=1}^\infty \ln(r)x^r $$ My attempt Consider the following: (1) $$ J'(x) = \...
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0answers
13 views

Analytic continuation and complex integration of one variable of a multivariate function

Consider a function $f:(x_0,x_1,...,x_n)\in\mathbb{R}^n\rightarrow f(x_0,...,x_n)\in\mathbb{R}$. $f$ is a ratio of polynomials in $x_0,...,x_n$ which only has simple poles in the variable $x_0$, whose ...
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0answers
38 views

Proof of existence of maximal analytic continuation of a holomorphic germ

The following is from Lectures on Riemann Surfaces by O. Forster: 7.8. Theorem. Suppose $X$ is a Riemann surface, $a\in X$ and $\varphi\in\mathcal{O}_a$ is a holomorphic function germ at the point $...
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0answers
23 views

Correct way to analytically continue a multi-dimensional integral

Consider a multi-dimensional integral \begin{equation} \int dx_1 \int dx_2 ... \int dx_n f(x_1,...,x_n) . \end{equation} where $f$ has simple poles in each of the variables $x_1,...,x_n$. Is it ...
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0answers
33 views

Properties of Complex Function $f(z)=e^{-\frac{1}{z^{1/3}}}$

This post will be about a part of an example from my complex analysis book. Problem: They claim that there exist a function $J_+(z)$ holomorphic in the upper half plane $\operatorname{im}z>0$, ...
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0answers
149 views

What is an example of analytic continuation?

I’ve always heard of the idea of analytic continuation in th context of complex analysis, but what is one example that I could understand? If you could, please give an example that a Precalculus ...
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0answers
42 views

Extending Lacunary Series beyond their disks

I've been for the past year and half fascinated by the lacunary series $f(z) = \sum_{n=0}^{\infty} z^{2^n}$. This function obeys the following equation inside the unit disk. $$f(z^2) = f(z)-z$$ And ...
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0answers
32 views

Reason for choosing a particular analytical continuation of the factorial

From this answer I know the choice of continuous extention $\ \Gamma(z) = \int_{0}^{\infty}t^{z-1}e^{-t}dt\ $ is not unqiue. But is that particular extension the unique best choice in some sense? E.g....
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0answers
40 views

Extensions of real analytic functions with multiple variables

I wonder if my statements below are correct. Let $V$ be an open domain in $\mathbb{R}^d$, and $U$ an open domain in $\mathbb{C}^d$ with $V=\operatorname{Re}U:=\{\operatorname{Re}z:z\in U\}$. I want ...
1
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0answers
53 views

About the radius of convergence of power series representation of analytic functions

I had read that if $g$ is an entire function then the radius of convergence of any power series expansion of $g$ is infinite, because we can enlarge the radius of the disc where the Taylor series ...
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0answers
45 views

Analytic Continuation on the Unit Disk

I encountered a question in complex analysis. Given an analytic function on the complex plane, except for the positive real ray, and continuous everywhere - show it is entire. My idea is to use ...
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0answers
76 views

Uniqueness of rational functions

Suppose we have a rational complex function: $$G(z)=\frac{a_n z^n+a_{n-1}z^{n-1}+\dots+a_1 z+a_0}{b_m z^m+b_{m-1}z^{m-1}+\dots+b_1 z+b_0}$$ And now we consider only its evaluation on the complex ...
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0answers
59 views

Power series of a quaternionic variable q: what is the behavior on the boundary of convergence?

Suppose you have a power series $S(-)$ in a quaternion $q$, and the coefficients in S are complex numbers, and the radius of convergence is $1.$ Then the boundary of the region of convergence is the ...
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0answers
24 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 analytical form for $...
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0answers
115 views

Abel-Plana Analytic Continuation of the Partial Sums of the Zeta Function

So I am trying to analytically continue the formula for the partial sums of the Riemann zeta function by using the fact that, if $\zeta_{k}$ represents the $k^{\text{th}}$ partial sum of the zeta ...
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0answers
40 views

Analytic continuation for $\sum_{n=0}^{\infty}(\sqrt n+1/3)^{-s}$

Define a function $F(s)$ by: $$F(s)=\sum_{n=0}^{\infty}(\sqrt n+1/3)^{-s}$$ Is there a closed form expression for the analytic continuation of $F(s)$ to $F(-s)$?
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0answers
42 views

Analytic Continuation of Sum $ \sum_{n=0}^{\infty} e^{-b \sqrt n}$

Suppose we have the following function: $$ f(b)=\sum_{n=0}^{\infty} e^{-b \sqrt n}$$ Is there a closed form expression for the analytic continuation of $f(b)$ to $f(-b)$?
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0answers
40 views

Natural Boundary of Sum $\sum_{n=0}^{\infty} e^{-b n^p}$

Is it possible to prove for which $p$ does the sum $$\sum_{n=0}^{\infty} e^{-b n^p}$$ have a natural boundary on the imaginary b axis?
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0answers
57 views

Is it possible for distinct geodesics to be equivalent over a finite segment?

Is it possible for two geodesics $\gamma_1, \gamma_2$ to be identical within a finite interval without being identical outside the interval? IOW: $\gamma_1(t) = \gamma_2(t)$ for $t \in (A,B)$ but $\...
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0answers
33 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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0answers
61 views

A shortcut for analytic continuation?

Let $P(x)$ be a nonconstant integer polynomial with nonnegative coëfficiënts such that the equation $y= P(y)$ has only one real solution $q$. Let $x_1=P(0)$ and $x_n = P(x_{n-1})$. $$f(z) = \sum_{n&...
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0answers
88 views

Analytic Continuation with Real Coefficients

I have an interesting problem that I cannot make progress on over several days. It is as follows: Suppose $f(z) = \sum a_nz^n$ is analytic around $0$ such that $a_n \geq 0$ for all $n$, in ...
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0answers
73 views

Analytic continuation vs. series convergence near convergence boundary

Citing Wikipedia, the Riemann zeta function is the analytic continuation of $$ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} $$ The series itself is only convergent in the right half complex plane ...
0
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0answers
28 views

Clarification of the proof of Euler's identity regarding the generating function for partitions.

In reference to this question which I asked here couple of days back but didn't get any answer I am posting this question to clarify whether we can able to extend Euler's identity regarding the ...
0
votes
0answers
11 views

Analytic continuation of circular orbits in restricted three body problem

I don't understand a Birkhoff's construction for circular restricted three body problem. "The x,y-plane is rotating. Consider the orbits belonging to the given value of C(Jacobi's constant) which ...
0
votes
0answers
14 views

Understanding analytic continuation and swapping integration and analytic continuation.

This question is partly Physics motivated, I am trying to study real-time dynamics of a system, but the theory is well-defined in Euclidean space-time. I have a complicated integrand, function of ...