Questions tagged [analytic-continuation]

For questions related to analytic continuation

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11 views

Can $f(z)=\int_E \frac{dt}{t-z}$ be extended to an entire function?

Suppose that $E \subset \mathbb{R}$ is compact and $m(E)>0$. Let $\Omega=\mathbb{C} \setminus E$ and \begin{equation} f(z)=\int_E \frac{dt}{t-z} \end{equation} for all $z \in \Omega$. I think that ...
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19 views

Extension of $f\in\mathcal{H}(\mathbb{D})$ that maps a circular arc to an analytic curve

Let $f\in\mathcal{H}(\mathbb{D})$ (i.e. it is a holomorphic function on the open unit disc) and $\gamma$ a (open) circular arc $\in \partial\mathbb{D}$ such that $f\in\mathcal{C}^0(\mathbb{D}\cup\...
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21 views

Analytic continuation of a holomorphic function $f$ is the same along any closed curve in $D$ implies $f$ extends to $D$?

Let $D\subset\mathbb{C}$ be a domain and let $f$ be the germ of a holomorphic function which admits unrestricted analytic continuation in some $E$ that contains $D$. Furthermore, suppose that analytic ...
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55 views

Why is analytic continuation practical?

I was exploring the Riemann Zetta function, and I observed that $\zeta (s)$ is not normally defined for $s $ such that $\Re (s) \leq 1$, but analytically continued to the whole complex plane. And ...
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27 views

analytic continuation of $\Phi(s)=\sum e^{-n^s}?$ [duplicate]

How do you analytically continue $\Phi(s)=\sum e^{-n^s}?$ I'm auditing complex analysis next fall but I'm curious about this function. I'm interested in this function because it converges and I ...
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22 views

Extending Local Function Germ to Global Morphism

I am wondering if the following argument is true. Let $f\colon (\mathbb{F}^n,0)\to (\mathbb{F}^p,0)$ be a function germ, here $\mathbb{F}$ is either $\mathbb{R}$ or $\mathbb{C}$. For any pair of $\...
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59 views

Analytic continuation of $H_x^{(k)}=\sum_{n=1}^x \frac{1}{n^k}$?

Is there an analytic continuation of the generalised harmonic number $H_x^{(k)}=\sum_{n=1}^x \frac{1}{n^k}$ to the positive reals x, for $k>1$? I can’t find anything useful through Google, just ...
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34 views

Is $\Phi$ the analytic continuation of $\Psi?$

Is $$\Phi(s)=\sum_n e^{-(n^s)}$$ the analytic continuation of $$\Psi(s)=\sum_n (e^{-n^{-s}}-1)=\sum_{k\ge 1} \frac{(-1)^k}{k!} \zeta(sk)$$ which is analytic on $\Bbb C^∗−∪1/k$ because $(s−1)ζ(s)=O(...
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PDE and continuation method / Newton's method

I want to solve the nonlinear elliptic equation $$ \begin{cases} \Delta u+ f(u)=g \;\; \text{in} \; \; (0,1)\\ u(0)=u(1)=0 \end{cases} $$ where $f\in C^1(\mathbb{R})$ and $g\in L^2(0,1)$ My idea, ...
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125 views

Analytic continuation of $\Phi(s)=\sum_{n \ge 1} e^{-n^s}$

While discussing theta functions, I thought: $\zeta(s)=\sum n^{-s}=1+2^{-s}+3^{-s}+ \cdot\cdot\cdot$ $\Phi(s)=\sum e^{-(n^s)}=e^{-1}+e^{-(2^s)}+e^{(-3^s)}+\cdot\cdot\cdot $ What is the analytic ...
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42 views

How to show that the limit function fullfills $|F(z)| = 1$ for $|z|=1$ using the maximum principle?

Let $D\subset \mathbb{C}$ be bounded by $|z| = 1$ and some continuum $C$ in $|z| < 1$ such that $0 \not \in D$. The sequence $(f_n)$ of holomorphic and injective functions is defined on $D$ and ...
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21 views

The Definition of Analytic Continuation

I’ve been reading through Section 9.2 in Conways book on complex functions and I am a bit stuck on understanding Conway’s remark in the second to last paragraph on p.214. To be precise, “That is, $$...
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108 views

Maximal extension of domain of $f(x)=\sum_{n=1}^\infty n^{\frac{1}{\log_n(x)}}$ using analytic continuation

Given $$ f(x)=\sum_{n=1}^\infty n^{\frac{1}{\log_n(x)}}=\sum_{n=1}^\infty e^{\frac{\ln^2(n)}{\ln(x)}} .$$ By inspection this is a sum of nonlinear hyperbolas, over $n$. It's because $\ln(x)\ln(y)=\ln^...
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3answers
114 views

Analytic continuation of a conformal map across the unit circle

I know that if $f$ is a conformal mapping of $\mathbb{D}$ onto some domain $D$ such that $\partial D$ is a Jordan curve, then $f$ has a continuous extension up to $\partial \mathbb{D}$ such that $f(\...
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37 views

Clarifications on a proof on analytic continuations

In the book Function of one complex variable of John B. Conway, the proof of this lemma on analytic continuations : Lemma on Analytic Continuations rest of the proof : Proof of the preceding lemma ...
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Deducing domain of analyticity of the Laplace transform from a big O estimate: possible or not?

In e.g. the field of analytic number theory it seems quite normal to use transforms (for example the Laplace) to deduce results about asymptotics for certain counting functions, like in the prime ...
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32 views

On the continuation of the gamma function, $\Gamma(x)$ [duplicate]

So, on Wikipedia, it states that the gamma function, $\Gamma(z)$, $z\in\mathbb{C}$ and $\Re(z)\notin S$, for $S=\lbrace0,-1,-2,\cdots\rbrace$, is the analytic continuation of the integral $$ I(z)=\...
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75 views

How to numerically perform analytic continuation?

I understand that one can in theory analytically continue a function by repeatedly computing new Taylor series. Suppose for example we have an analytic function $f$ defined on some open set $U$ and ...
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Analysis of the Adjoint representation of Lie Algebra

I'm attempting to find $f\left(ad_{x}\right)y$ where $ad_{x}y=xy-yx$ for any function f and rings x,y. adjoint representation I have a way to calculate it for any holomorphic function. However, I'm ...
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29 views

Analytic Continuation of Fractional Derivatives

I've been doing some work with fractional calculus, and I have been running into a problem. For a given meromorphic function $f: \mathbb{C}\to \mathbb{C}$, if a formula of the form $$(D^{\alpha}f)(0)=...
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68 views

On analytic continuation

If we have a power series, can we always get an analytic continuation of it? And if not, when can we? And is there a simple general method to get an analytic continuation of a divergent power series?
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91 views

How can one get an analytic continuation of a power series?

What method should one use to get an analytic continuation of this power series: $$\sum_{n=0}^\infty \frac{x^n}{n\#}$$ ; where $n\#$ is the product of all primes less or equal to $n$. The radius of ...
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148 views

$f(z) = z + f(z^2)$ outside the unit disk?

The function $$ f(z) = \sum_{n=0}^\infty z^{2^n} $$ which satisfies the functional equation $f(z) = z + f(z^2)$ is a classic example of a function analytic in $\mathbb{D} = \{z:|z|<1\}$ that cannot ...
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18 views

Analytic continuation of a multivariable function composition

Let $f\colon \mathbb{R}^n\to\mathbb{R}$ and, for all $i=1,\dots,n$, let $g_i\colon\mathbb{R}\to\mathbb{R}$. Then $f(g_1(x),\dots,g_n(x))$ maps $\mathbb{R}\to\mathbb{R}$. Suppose that each $g_i$ has ...
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39 views

Do all values of $\zeta (s)$ follow from the Dirichlet series definition of $\zeta (s)$?

If we define $\zeta (s)$ by $$\zeta (s)\overset{\text{def}}{=}\sum_{n\gt 0}n^{-s},\, \operatorname{Re}s\gt 1,$$ does it follow from the definition that e.g. $\zeta (-1)=-\frac{1}{12}$? Or is it ...
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39 views

Meromorphic continuation of the Dedekind zeta function $\zeta_K(s)$ for all $s$ with $\sigma > 1 - \frac{1}{d}$.

Let $K$ be a number field of degree $n$ and $\mathcal{O}_K$ denote the ring of integers of $K$. For any complex number $s=\sigma +it$ with $\sigma > 1$, we define the Dedekind zeta function by the ...
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40 views

Meromorphic continuation of $\zeta(s)$ for all $s \in \mathbb{C}$ with $Re(s) > 0$

I am trying to obtain the meromorphic continuation of the Riemann zeta function $\zeta(s)$ for all $s \in \mathbb{C}$ with $Re(s) > 0$. Using the Abel's summation formula I've obtained that $$ \...
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60 views

The Riemann zeta function and analytic continuation, how many solutions are possible to his analytic continuation?

When Riemann developed the zeta function in the complex plane, he used a process called analytic continuation. He found only one solution. Does this mean that there was only one analytic solution to ...
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1answer
47 views

On a complex function $f(z)$ which is $\sqrt{z^2-1}\in\Bbb R $ when $x>1$

Let $f(z)$ be a complex function whose value is $\sqrt{z^2-1}\in\Bbb R$ when $z\in \Bbb R$ is $>1$, and let's suppose $f(z)$ is holomorphic if $1<|z|<\infty$. I want to integrate $f(z)$ over $C$, a ...
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1answer
54 views

How to analytically continue the ‘two-dimension’ zeta function?

How to meromorphically continue $$f(s)=\sum^\infty_{n=1}\sum^\infty_{m=1}\frac1{(an+bm)^s}$$ to $\Re s\le 2$? ($a,b$ are positive constants.) In the case $a=b$, we have $$f(s)=a^{-s} \sum^\infty_{n=1}...
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28 views

Asking for a reference which proves analytic continuation of Hypergeometric Function

I am self studying a research paper in analytic number theory and it requires analyticity of Hypergeometric Function . I searched google but i couldnot found anything precize on how to prove ...
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25 views

What is the meaning of analytic continuation of $\Psi(z)$ when $z$ crosses the positive real axis from above?

I am reading an article A class of analytic perturbations for one-body Schrödinger Hamiltonians. In the text there are expression like this $...\Psi(z)$ can be analytically continued when z crosses ...
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Doubt regarding proving analyticity of a power series

I am self studying concepts of complex analysis from Ponnusamy and Silvermann "Complex Variables with Applications" In the chapter of analytic continuation in basic concepts authors mention that ...
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43 views

Properties of $\sum_{n=1}^{\infty} n^{-s\log n}$

I want to know about the function $\displaystyle f(s):=\sum_{n=1}^{\infty} n^{-s\log n}$. In particular, can this function be continued analytically around $s=0$?
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41 views

Do Divergent Integrals have a unique regularisation?

I know that the same question for divergent sums is false, but cannot find much on divergent integrals. For example, consider the following divergent integral for positive reals $a,b$: $\...
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141 views

A method to iterate the exponential function a non-integer number of times?

Notation We employ the following notation: $$ a_1 = e^{x} $$ $$ a_2 = e^{e^{x}} $$ $$ a_3 = e^{e^{e^{x}}} $$ $$ a_n = e^{\vdots^{e^{x}}}$$ We also use define $c$ by: $$ e^c =c$$ Motivation ...
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1answer
71 views

Regarding applying concept of analytic continuation in Analytic number theory

I am studying analytic number theory from Tom M Apostol introduction to analytic number theory, Actually Concept of analytic continuation was not taught by my instructor who took complex analysis ...
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Verifying uniqueness of my tetration

Previous two posts: Numerical instability of an extended tetration Verifying tetration properties Update: The first link only verifies continuity on $\mathbb R$, and so continuity cannot be used for ...
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1answer
88 views

Let $f$ be a function which is continuous on the closed unit disc and analytic on the open disc.

Let $f$ be a function which is continuous on the closed unit disc and analytic on the open disc. Assume that $|f(z)| = 1$ whenever $|z| = 1$. Show that the function $f$ can be extended meromorphically ...
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Reference request for a book that covers analytic continuation in great detail starting from basics

I have earlier self studied Tom M Apostol Introduction to analytic number theory after doing a course in complex analysis, but my instructor at university didn't even mentioned analytic continuation, ...
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1answer
123 views

Continuation of functions beyond natural boundaries

The article Continuation of functions beyond natural boundaries by John L. Gammel states I am particularly interested in the convergence of the $[N/N+1]$ Padé approximants beyond the natural ...
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1answer
60 views

Are all analytic continuations consistent with $1-1+1-1+\dots = \frac{1}{2}$?

Background: This is a follow-up to this question, which is the same question except for a different series (and I suspect the answer may be different for this series—more on that later). In short, I ...
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75 views

Are all analytic continuations consistent with $1+2+3+\dots=-\frac{1}{12}$? [duplicate]

Background: One way to "prove" that $1+2+3+\dots=-\frac{1}{12}$ is to define the function $f(z)=\sum_{n=0}^\infty \frac{1}{n^z}$ for $z\in\mathbb C, \operatorname{Re}(z)>1,$ then define $\zeta(z)$ ...
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59 views

Is it possible to deduce that $\mathbb C^2 \setminus \{ (0,0) \}$ is not affine directly from Hartogs's principle?

It is well-known that $X=\mathbb C^2 \setminus \{ (0,0) \}$ is not an affine variety. Perhaps the simplest proof of this fact consists of showing that every regular function on $X$ extends to a ...
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Integral involving a Gaussian and a fraction.

This question is a generalization of An identity involving the incomplete Beta function. . Let $x\ge 0$ and $\epsilon_\pm \in (1,\infty)$. We consider the following integral: \begin{equation} {\...
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329 views

Is the “sum of all natural numbers” unique?

A while ago, there was a great hype about the “identity” $$\sum_{n=1}^{\infty} n = -\frac{1}{12}.$$ Apart from some series manipulations where the validity seems to be at least questionable, the ...
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1answer
60 views

Analytic continuation in a proof in Apostol's analytic number theory textbook

In Apostol's Introduction to Analytic Number Theory, the following equation is derived for real $s>1$: $$\Gamma(s)\zeta(s)=\int_0^{\infty}\frac{x^{s-1}}{e^x-1}\,\mathrm{d}x$$ Then, the fact that ...
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35 views

Proof of nonexistence of holomorphic continuation.

Let $D \subseteq \mathbb{C}$ be a domain and $\emptyset \neq U \subsetneq D$ open. I want to show the existence of a holomorphic function $f: U \rightarrow \mathbb{C}$ without holomorphic continuation ...
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15 views

Is a solution of an elliptic PDE constant if all of its derivatives vanish at a point? (in analogy to unique continuation principle)

Consider a one-dimensional elliptic second order differential equation on an interval $[a, b]$. I am specifically interested in Sturm-Liouville problems where the principal symbol is the Laplacian and ...
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1answer
59 views

Analytic continuation of geometric series using Taylor expansion

I am currently trying to get a better grasp of the concept of analytic continuation and so I am working through this example: Let $$f:(-1,1)\to\mathbb{R}~,~x\mapsto\sum_{k=0}^\infty x^k = \frac{1}{...

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