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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1answer
30 views
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,
$$
\int_0^{\infty} dx\; x^{-2g} e^{-x^2}= \Gamma\...
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0answers
21 views
Principle of analytic continuation
I've read and understood the proof for the complex plane for the analytic continuation theorem. The proof relies on $E$ being both closed and open, with $\quad E = \bigcap _ { n \geq 0 } E _ { n }$ ...
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Bessel function of 1st kind and integer order in complex plane.
Read the definition of Bessel function here. https://en.m.wikipedia.org/wiki/Bessel_function
Let me write the definition Bessel function $J_n(x)$ of 1st kind and integer order n,
$$J_n(x)=1/(2\pi) \...
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0answers
18 views
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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1answer
42 views
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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1answer
31 views
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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1answer
31 views
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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1answer
21 views
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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0answers
36 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} = ...
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27 views
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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0answers
24 views
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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2answers
319 views
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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48 views
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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30 views
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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0answers
203 views
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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1answer
48 views
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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1answer
25 views
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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0answers
50 views
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 ...
3
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1answer
66 views
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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27 views
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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0answers
25 views
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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0answers
26 views
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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0answers
30 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$, ...
1
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1answer
49 views
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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1answer
35 views
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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93 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
37 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$$
And ...
1
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1answer
49 views
Analytic continuation of a series raised to a power raised to a power?
Background
I recently realized I could construct the below formula:
$$ \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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1answer
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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}...
1
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1answer
50 views
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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0answers
26 views
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
$$
G(z)=\sum_{n=0}^{\infty} ...
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2answers
65 views
General form of the $n^{\text{th}}$ derivative of $x^x$
Can someone help me to confirm this identity that I have established, I really have no idea how I would go about proving that this is true.
By the way, this is not a homework assignment, I am just ...
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1answer
29 views
Proving a holomorphic function is identically zero when its zeros form a sequence that converge in some region
Before I jump in, just want to quickly note that this post exists already, I am using the same book, and I had a very similar confusion which led me to trying to prove the theorem in a slightly ...
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0answers
38 views
Is there a term for extensions of functions on discrete sets to continuous sets?
The pi function, $\Pi(z)$ – defined as $\Pi(z) = \Gamma(z+1)$, where $\Gamma(z)$ is the gamma function – extends the factorial in that
$$\Pi(n) = (n)!$$
for all positive integers $n$. In other words,...
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0answers
69 views
Analytic continuation of a holomorphic function in some domain
Quite generally I thought that every holomorphic function defined in some domain can be analytically continued into the entire complex plane. However the dedekind eta function seems to not have such ...
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1answer
53 views
For given problem if we change the setting what will happen?
I encountered following problem
and I solved it by using the hint provided. Thinking of it I noticed
that I am able to solve it even if I use the following function:
$$
F(z)=1/f(1/z)),\quad |z|> ...
0
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1answer
36 views
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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1answer
97 views
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 ...
0
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1answer
35 views
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 $$\ln2-\frac{1-z}{2}-\frac{(1-z)^2}{2\cdot 2^2}-\...
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0answers
30 views
Reason for choosing a particular analytical continuation of the factorial
From this answer I know the choice of continuous extention $\ \Gamma(z) = \int_{0}^{\infty}t^{z-1}e^{-t}dt\ $ is not unqiue.
But is that particular extension the unique best choice in some sense? E.g....
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0answers
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Dirac delta from poles of a function
Suppose we are given the simple expression
$$
F(k) = \frac{1}{E^2-E(k)^2}
$$
which has a pole when $E^2 = E(k)^2$ and where $E, E(k)$ are real numbers. When working with this expression (e.g. inside ...
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0answers
20 views
Relation between infinite product and regularized product
For a positive sequence $0\le\lambda_{1}\le\lambda_{2}\le\cdots$, consider an infinite product
\begin{equation*}
\prod_{i=1}^{\infty}\lambda_{i}:=\lim_{n\rightarrow\infty}\prod_{i=1}^{n}\lambda_{i}...
2
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1answer
76 views
Analytic continuation of several complex variables
Let $f(w_1,\ldots,w_n;z)$ be a holomorphic function of $n+1$ variables.
For every fixed $w_1\ldots w_n$, let $g(w_1,\ldots,w_n;z)$ be an analytic continuation of $f$ as a holomorphic function of $z$.
...
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0answers
30 views
Analytic continuation of Riemann Theta function to positive semidefinite matrices
The Riemann $\Theta$-function is defined as
$$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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0answers
36 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 ...
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2answers
48 views
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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1answer
30 views
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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0answers
48 views
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 $\gamma$...
3
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1answer
54 views
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}{...
