Questions tagged [analytic-continuation]
For questions related to analytic continuation
3
votes
0answers
16 views
Meromorphic continuation of the multifactorial
The double factorial $z!!$, like the normal factorial function, can be extended to the complex plane using
$$ z!! = 2^{z/2} \left(\frac{\pi}{2}\right)^{(\cos\pi z-1)/4} \left(\frac{z}{2}\right)! $$
...
0
votes
0answers
66 views
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....
1
vote
0answers
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 ...
1
vote
1answer
31 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 ...
3
votes
0answers
67 views
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 ...
0
votes
0answers
32 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 ...
0
votes
0answers
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 ...
0
votes
0answers
27 views
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.
0
votes
1answer
44 views
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 $z=x \in \mathbb{R}$?
1
vote
1answer
35 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,
$$
\int_0^{\infty} dx\; x^{-2g} e^{-x^2}= \Gamma\...
0
votes
0answers
31 views
Principle of analytic continuation
I've read and understood the proof for the complex plane for the analytic continuation theorem. The proof relies on $E$ being both closed and open, with $\quad E = \bigcap _ { n \geq 0 } E _ { n }$ ...
0
votes
0answers
18 views
Bessel function of 1st kind and integer order in complex plane.
Read the definition of Bessel function here. https://en.m.wikipedia.org/wiki/Bessel_function
Let me write the definition Bessel function $J_n(x)$ of 1st kind and integer order n,
$$J_n(x)=1/(2\pi) \...
0
votes
0answers
20 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 ...
1
vote
1answer
48 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 ...
0
votes
1answer
40 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 ...
0
votes
1answer
32 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:\...
0
votes
1answer
23 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 ...
2
votes
0answers
43 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} = ...
0
votes
0answers
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 ...
0
votes
0answers
28 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 ...
6
votes
2answers
346 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.
...
0
votes
0answers
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 ...
0
votes
0answers
37 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?
2
votes
0answers
293 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 ...
0
votes
1answer
79 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 ...
1
vote
1answer
34 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 ...
0
votes
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 ...
3
votes
1answer
85 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 ...
0
votes
0answers
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 ...
0
votes
0answers
27 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} \...
0
votes
0answers
30 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. $...
1
vote
0answers
32 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
vote
1answer
64 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 ...
0
votes
1answer
47 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 ...
1
vote
0answers
121 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 ...
1
vote
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$$
And ...
1
vote
1answer
50 views
Analytic continuation of a series raised to a power raised to a power?
Background
I recently realized I could construct the below formula:
$$ \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})...
1
vote
1answer
47 views
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
vote
1answer
52 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 $s\in[0,\infty)$.
0
votes
0answers
29 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
$$
G(z)=\sum_{n=0}^{\infty} ...
1
vote
2answers
65 views
General form of the $n^{\text{th}}$ derivative of $x^x$
Can someone help me to confirm this identity that I have established, I really have no idea how I would go about proving that this is true.
By the way, this is not a homework assignment, I am just ...
0
votes
1answer
45 views
Proving a holomorphic function is identically zero when its zeros form a sequence that converge in some region
Before I jump in, just want to quickly note that this post exists already, I am using the same book, and I had a very similar confusion which led me to trying to prove the theorem in a slightly ...
0
votes
0answers
41 views
Is there a term for extensions of functions on discrete sets to continuous sets?
The pi function, $\Pi(z)$ – defined as $\Pi(z) = \Gamma(z+1)$, where $\Gamma(z)$ is the gamma function – extends the factorial in that
$$\Pi(n) = (n)!$$
for all positive integers $n$. In other words,...
0
votes
0answers
77 views
Analytic continuation of a holomorphic function in some domain
Quite generally I thought that every holomorphic function defined in some domain can be analytically continued into the entire complex plane. However the dedekind eta function seems to not have such ...
4
votes
1answer
54 views
For given problem if we change the setting what will happen?
I encountered following problem
and I solved it by using the hint provided. Thinking of it I noticed
that I am able to solve it even if I use the following function:
$$
F(z)=1/f(1/z)),\quad |z|> ...
0
votes
1answer
42 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 ...
0
votes
1answer
148 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
votes
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 $$\ln2-\frac{1-z}{2}-\frac{(1-z)^2}{2\cdot 2^2}-\...
1
vote
0answers
32 views
Reason for choosing a particular analytical continuation of the factorial
From this answer I know the choice of continuous extention $\ \Gamma(z) = \int_{0}^{\infty}t^{z-1}e^{-t}dt\ $ is not unqiue.
But is that particular extension the unique best choice in some sense? E.g....
3
votes
0answers
35 views
Dirac delta from poles of a function
Suppose we are given the simple expression
$$
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 ...
