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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2answers
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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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0answers
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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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2answers
41 views
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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1answer
31 views
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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0answers
37 views
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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23 views
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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1answer
32 views
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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0answers
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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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36 views
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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30 views
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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1answer
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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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1answer
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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,
$$
\int_0^{\infty} dx\; x^{-2g} e^{-x^2}= \Gamma\...
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0answers
40 views
Principle of analytic continuation
I've read and understood the proof for the complex plane for the analytic continuation theorem. The proof relies on $E$ being both closed and open, with $\quad E = \bigcap _ { n \geq 0 } E _ { n }$ ...
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Bessel function of 1st kind and integer order in complex plane.
Read the definition of Bessel function here. https://en.m.wikipedia.org/wiki/Bessel_function
Let me write the definition Bessel function $J_n(x)$ of 1st kind and integer order n,
$$J_n(x)=1/(2\pi) \...
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21 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
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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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1answer
42 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
34 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
25 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
44 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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29 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
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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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2answers
368 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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51 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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44 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
333 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
91 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
45 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
51 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 ...
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1answer
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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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28 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
31 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
34 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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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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1answer
66 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
50 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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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
42 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$$
And ...
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1answer
50 views
Analytic continuation of a series raised to a power raised to a power?
Background
I recently realized I could construct the below formula:
$$ \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}...
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1answer
56 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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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} ...
