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Questions tagged [analytic-continuation]

For questions related to analytic continuation

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Real world applications of Riemann surfaces of holomorphic functions [on hold]

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}{...
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Identity Theorem: Extending from $\mathbb{R}$ to $\mathbb{C}$

Suppose we have $f_1, f_2: \mathbb{C} \rightarrow \mathbb{C}$ holomorphic, and $f_1 = f_2$ on $\mathbb{R}$. Can we then say $f_1 = f_2$ identically on $\mathbb{C}$? This appears to be true by the ...
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Proving existence of a function with no analytic extension beyond any boundary point of a given region

The following is an old qualifying exam question in Complex Analysis. Let $G$ be the open strip $\{z \in \Bbb C | 1 < \Im(z) < 2\}$. Prove that there exists a function $f$ that is analytic ...
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Why are these two functions different for negative numbers?

Looking at some straightforward generalizations of the Sophomore's Dream identity I derived the following identity which works for $\Re(b) > 0$: $$ \int_0^1 x^{a x^b} dx = \sum_{n=0}^{\infty} \frac{...
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Hanging picture on the wall with two nails

There's picture which we have to hang on the wall with two nails. The two ends of a string are attached to both of the upper edges of the picture. a) How do we have to hang the picture if it falls (...
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Is there an analytically continued function of $z^p$ at zero?

For a rational number $p > 1$. We know that the function $z^p$ is holomorphic on $\mathbb{C} \setminus \mathbb{R}^-$ (excluding $z = 0$). Is there an analytic continuation of the function $z^p$ at ...
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Extending $\zeta$ to a meromorphic function on $\mathbb{C} - \{1\}$

I know that we can extend $\zeta (s)$, originally defined on $\Re(s)>1$ by the sum $\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}$, to the domain $\mathbb{C} - \mathbb{Z}$ by this definition: $$\zeta(s)...
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How can I know whether an analytic function is analytically continuable or not along a path? [closed]

How can I know whether an analytic function is analytically continuable or not along a path ?
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How is extension by analytic continuation done?

I understand that by defining an expression for a function to be analytic, we can extend the range of the expression beyond it's usual range. One case is that a series which is asymptotic to a ...
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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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analytic continuation over a manifold of arbitrary genus

I am trying to understand the interplay between global topology and analytic continuation. Suppose $M$ is a connected analytic manifold with metric $g$ which is everywhere analytic. Define $\gamma(g(...
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complex analysis- simple poles of the gamma function

I know that the $\Gamma(z)$ has simple poles at $0, -1, -2,...$ Does that mean $\Gamma(z/2)$ has simples pole at $0, -1/2,-1,...$? Also, $Res(\Gamma,-k)=(-1)^k/k!$, so is the residue of $\Gamma(z/2)$...
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Translation of integration contour

I have an argument where, with a meromorphic function $\phi$ satisfying $\phi(s) = \phi(1-s)$, the following equality appears: $$\int_{(2)} \phi(s) y^{-s} \, ds = \int_{(1/2)} \phi(s) y^{-s} \, ds$$ ...
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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 ...
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On Alexander polynomial of a knot

The Alexander polynomial of a knot is of the form $$\Delta(t)=det(V^T-tV),$$ where $V$ is the Seifert matrix, see http://archive.lib.msu.edu/crcmath/math/math/a/a116.htm. What is geometric or some ...
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Does a branch cut discontinuity determine a function near the branch point?

Suppose $g(z)$ is analytic on a disc centered at the origin, except along the negative real axis where it has a branch cut discontinuity. Also assume that $g(0)=0$. Let $h(x)$ give the discontinuity ...