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Questions tagged [analytic-continuation]

For questions related to analytic continuation

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Analytic continuation of a holomorphic function bounded on neighbourhood of 'small' compact subset $K\subset\mathbb{C}$

I have two questions regarding the following exercise. First, is my solution (see below) correct? Second, how would the solution go that the writer of the book had in mind (i.e. using the hint)? ...
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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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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 ...
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Deriving Analytic Continuation of Riemann Zeta Function [duplicate]

I'm trying to derive the formula for analytic continuation of riemann zeta function but I can't find a way. How do we find results like Zeta(-1)= -1/12 ?
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Asymptotic expansion of $Li^{-1}$ and zeros of $F(s)$ and $G(s)$

If you downvote please leave some constructive feedback. I would like to compare and visualize/gain insight about the zeros of two functions, $F(s)$ and $G(s).$ $\pi(m)$ is the prime counting ...
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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 ...
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Why is the borel sum analytic

I am currently reading in a book about Borel sums as a method of analytic continuation of power series. So given a power series $\sum_{n=0}^{\infty}a_nz^n$ the borel sum is defined as $\int_0^\infty e^...
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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 ...
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Is the analytic continuation of an even function even?

Suppose we have an even function, $f$, defined on the real line. Suppose this function admits an analytic continuation, defined on the whole of $\mathbb{C}$. Does this imply that said analytic ...
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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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Meromorphic continuation of the multifactorial

The double factorial $z!!$, like the normal factorial function, can be extended to the complex plane using $$ z!! = 2^{z/2} \left(\frac{\pi}{2}\right)^{(\cos\pi z-1)/4} \left(\frac{z}{2}\right)! $$ ...
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Geometric view of analytic continuation

Can the shift of center of convergence for power series from point to point in a path of overlapping circles, in analytic continuation, be interpreted as a translation in any way? https://upload....
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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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Extend $f(z)=\frac{1}{z^n +z^{n-1}+…+z^2 + z^{-n}}+\frac{c}{z-1}$

find $c $ such that $ f(z)=\frac{1}{z^n +z^{n-1}+...+z^2 + z^{-n}}+\frac{c}{z-1}$ can be extended to be analytic at $z=1$ , when $n\in \mathbb{N}$ when $n$ is fixed. The given function I write it ...
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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 ...
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Relationship between cauchy principal value and integrability of a singular point

Under what conditions does an integral have a cauchy principal value and how is it related to an integral having an integrable singularity? E.g $$p.v \int_{-\delta}^{\delta} \frac{dz}{z} = 0.$$ If I ...
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Does analytic continuation give actual values

If analytic continuation gives the wrong answer sometimes, at least under typical/basic reasoning, like when it assigns a divergent series a finite value, then why do we trust it to give us values ...
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Quaternion algebra using analytic continuation

As for complex variables, do we use analytic continuation to find things like $sin(j)$, $i^k$, and so on? Is there another method or do these expressions even have values at all.
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Analytic continuation of $\sin(z)$ [closed]

Why $$\sin{ (z)} =\frac{e^{iz}-e^{-iz}}{2i}$$ the only analytic function, is equal to $\sin{(x)}$ for $z=x \in \mathbb{R}$?
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Analytic continuation of Gamma function and negative moments of normal distribution

I want to evaluate the divergent integral: $$\int_0^{\infty} dx\; x^{-2} e^{-x^2}$$ My plan is to calculate the following integral instead, $$ \int_0^{\infty} dx\; x^{-2g} e^{-x^2}= \Gamma\...
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Principle of analytic continuation

I've read and understood the proof for the complex plane for the analytic continuation theorem. The proof relies on $E$ being both closed and open, with $\quad E = \bigcap _ { n \geq 0 } E _ { n }$ ...
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Bessel function of 1st kind and integer order in complex plane.

Read the definition of Bessel function here. https://en.m.wikipedia.org/wiki/Bessel_function Let me write the definition Bessel function $J_n(x)$ of 1st kind and integer order n, $$J_n(x)=1/(2\pi) \...
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Analytic contiunation

this is more of a broader question. Say I have an analytic complex function $f$ that is defined on the open unit circle, but I know that the limit of $f$ when $|z|$ approaches 1 is 0. Can you define ...
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Simple case of analytic continuation

I have yet to formally study complex analysis, yet the topic of analytic continuation, specifically with respect to the Riemann Zeta function, is fascinating. My question is, what are some ...
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Analytic continuation of $\sum _{n=0}^{\infty} \frac{z^{2^n}} {2^n}$ beyond the unit disc

The function : $\sum _{n=0}^{\infty} \frac{z^{2^n}} {2^n}$ convergs to holomorphic function $f$ on $D_1(0)$ and is continious on $\overline{D_1(0)}$. I need to prove that f can't be exteneded to any ...
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Definition clarification of analytic continuation of holomorphic function

I was given the definition of anlytic continuation as in Stein's book- given two regions $\Omega\subset \Omega'$ for analytic functions $F:\Omega'\to \mathbb{C}$ is an analytic continuation of $f:\...
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Is there a “monotonicity” property for analytic continuation?

If I have two complex functions defined by power series $A(z) = \sum a_n z^n $, $B(z) = \sum b_n z^n$ with $|a_n| \ge |b_n|$ for all $n$, and I know that $A$ converges in some set $U_1$ and defines a ...
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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} = ...
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If a sequence of distinct points in a bounded connected open $\Omega$ doesn't converge in $\overline\Omega$, why can we apply analytic continuation?

Let $\gamma\subset\Bbb C$ be a closed, simple (if $\gamma:[a,b]\mapsto\Bbb C$, $\gamma$ is injective on $(a,b)$, so the curve doesn't intersect except for $\gamma(a)=\gamma(b)$) and piece-wise regular ...
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Meromorphic continuation of function analytic on the open unit disc [duplicate]

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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Analytic continuation of harmonic series

Is there an accepted analytic continuation of $\sum_{n=1}^m \frac{1}{n}$? Even a continuation to positive reals would be of interested, though negative and complex arguments would also be interesting. ...
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Maximal analytically continued domain

Can a power series or a Laurent series always be analytically continued into a domain strictly larger than its convergent annulus of finite radii? Is there a way to find its maximal domain that it can ...
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Applications of Riemann surfaces in engineering or physics

I know that the maximal analytic continuation of a holomorphic function is an example of Riemann surfaces but don't know what it is used for. What can we do with this surface?
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Real world applications of Riemann surfaces of holomorphic functions [closed]

The maximal analytic continuation of a holomorphic function is an example of Riemann surfaces. What is it used for? Please edit the question to limit it to a specific problem with enough detail ...
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Analytic continuation of the logarithm

This is an example from Serge Lang Complex Analysis book, it says Let us start with the function $log(z)$ defined by the usual power series on the disc $D_0$ which is centered at $1$ and has ...
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Holomorphic functions reflected through segments that aren't on the real axis

My question concerns using the Schwarz Reflection principle (or symmetry principle) to reflect regions in the domain of a holomorphic function into a symmetric (with respect to a line segment) region ...
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Maximal Natural Domain of An Analytic Function on Complex Plane

I want to ask a question of the maximal natural domain of an analytic function. We know that on $\mathbb{C}^n$, a domain $U$ is a domain of holomorphy if and only if $U$ is a pseudoconvex domain, and ...
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Is the Lambert W function analytic? If not everywhere then on what set is it analytic?

I would appreciate if someone can help me answer the following questions. Although I read several papers and documents on the Lambert W function, I could not assess on what set is this function (or ...
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Closed curve for an analityc function

If I have an analityc function $f(z)$ on a domain $B \subset \mathbb{C}$ and a simple and closed curve $C$ that encloses $B$, and if $|f|$ is constant over C, the $f$ is constant on $B$? I haven’t ...
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Generalizing rising and falling factorial to complex arguments preserving zeroes

Falling factorials count injective functions. There are no injective functions $ A \to B $ if $|A| \gt |B|$ . Is there a way to extend the definition of the falling factorial, ideally to $\mathbb{C} \...
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Can we evaluate noninteger hyperoparations?

The hyperoperation $a[n]b$ is addition when $n=1$, multiplication when $n=2$, exponentiation when $n=3$, tetration when $n=4$, and so on. What happens when $n$ is noninteger? Can we evaluate, e.g. $...
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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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Is analytic continuation well-defined as a summation method?

I am not well versed in summation methods or complex analysis, so I will be presenting a detailed view of my question with examples to illustrate my point as well as a few guiding questions that got ...
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Analytic continuation of quotient of analytic functions

Suppose $f(z)$ and $g(z)$ are defined for some open subset $U$ of the complex plane, and that they are holomorphic on that subset. We then know that their pointwise quotient $(f/g)(z)$ is meromorphic ...
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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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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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Analytic continuation of a series raised to a power raised to a power?

Background I recently realized I could construct the below formula: $$ \lim_{ x \to 1 }(1-x)(\sum_{r=1}^\infty b_r x^{r^\kappa} ) = (\sum_{\tilde r = 1}^\infty \frac{ b_\tilde r }{\tilde r ^\kappa})...
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What is this renormalization/continuation trick called and why does it work?

Consider the integral $$ I_n = \int_0^\infty x^n \sin(x) dx $$ for $n\in\mathbb R$. The integral exists and is finite for $-2 < n <0$, giving the value $I_n = \Gamma(n+1)\cos\left(\frac{n\pi}{2}...
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how to prove that a function vanishing at an interval is identically zero?

Let $\phi(s):=\int_{0}^{\infty}\exp(-st)g(t)dt$ for $g\in L_1(0,\infty)$. Assume that $\phi(s)=0$ for $s\in[0,\frac{1}{2})$. How to prove that $\phi(s)=0$ for every $s\in[0,\infty)$.
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Contour Deformation in the Laplace Inversion Formula

Following Szpankowski - Average Case Analysis of Algorithms on Sequences the exponential generating function of $g(n)$ which is thought to be analytic in $n$ is defined as $$ G(z)=\sum_{n=0}^{\infty} ...