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Questions tagged [analytic-continuation]

For questions related to analytic continuation

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Analytic continuation of $\sin(z)$ [on hold]

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} ...
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General form of the $n^{\text{th}}$ derivative of $x^x$

Can someone help me to confirm this identity that I have established, I really have no idea how I would go about proving that this is true. By the way, this is not a homework assignment, I am just ...
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Proving a holomorphic function is identically zero when its zeros form a sequence that converge in some region

Before I jump in, just want to quickly note that this post exists already, I am using the same book, and I had a very similar confusion which led me to trying to prove the theorem in a slightly ...
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Is there a term for extensions of functions on discrete sets to continuous sets?

The pi function, $\Pi(z)$ – defined as $\Pi(z) = \Gamma(z+1)$, where $\Gamma(z)$ is the gamma function – extends the factorial in that $$\Pi(n) = (n)!$$ for all positive integers $n$. In other words,...
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Analytic continuation of a holomorphic function in some domain

Quite generally I thought that every holomorphic function defined in some domain can be analytically continued into the entire complex plane. However the dedekind eta function seems to not have such ...
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For given problem if we change the setting what will happen?

I encountered following problem and I solved it by using the hint provided. Thinking of it I noticed that I am able to solve it even if I use the following function: $$ F(z)=1/f(1/z)),\quad |z|> ...
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Understanding Proof on Analytic Continuation .

In Book Complex Analysis by Stain and Shakarchi . I read following proof . I understand that for $f\neq 0$ then there exist some non zero m such that $a_m\neq 0$ but I do not understand why f(z) can ...
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Analytic continuation of the incomplete beta function

Is there a rigorous proof for the analytic continuation of the incomplete beta function $B(x;a,b)$ for all values of $a$ and $b$? The incomplete beta function normally restricts the values of $a,b$ as ...
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Prove that the function can be continued into a larger domain

Prove that the function $f(z)=\displaystyle\sum_{n=1}^{\infty}(-1)^{n+1}\frac{z^n}{n}$ can be continued into a larger domain by means of the series $$\ln2-\frac{1-z}{2}-\frac{(1-z)^2}{2\cdot 2^2}-\...
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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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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 ...
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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}...
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Analytic continuation of several complex variables

Let $f(w_1,\ldots,w_n;z)$ be a holomorphic function of $n+1$ variables. For every fixed $w_1\ldots w_n$, let $g(w_1,\ldots,w_n;z)$ be an analytic continuation of $f$ as a holomorphic function of $z$. ...
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Analytic continuation of Riemann Theta function to positive semidefinite matrices

The Riemann $\Theta$-function is defined as $$Θ(z|Ω)=\sum\limits_{q∈Z^N}e^{πiq⋅Ωq+2πiq⋅z},$$ where $\Omega$ has positive definite imaginary part to ensure convergence. In a particle physics ...
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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 ...
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When can one extend a function on the unit circle to an analytic function?

Suppose I have a function $f$ defined on the unit circle. Under what conditions can I define a new function $g$ defined on a subset of the complex plane containing the unit disk such that $f = g$ on ...
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Analytical continuation of $z^\alpha$

As $x^{\alpha}, \alpha\in\mathbb{R}$ is analytical in $(0,\infty)$, can I find an analytical continuation of $x^{\alpha}$, say $z^{\alpha}$ in $\mathbb{C}\backslash (-\infty,0]$?
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Analytic continuation along path

¡Hi! Anyone could give me a hint for this problem? Thanks. "Let $\gamma \colon [0,1] \to \mathbb{C}$ be a path and let $ \{(f_{t}, D_{t}) : 0 \leq t \leq 1 \}$ be analytic continuation along $\gamma$...
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Chain rule for higher-order derivatives

While Studying Chain rule In my Calculus Book It was written as :- $\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$ But, In the note it was mentioned that :- $\frac{d^2y}{dx^2} \ne \frac{d^2y}{...