Questions tagged [analytic-continuation]
For questions related to analytic continuation
398
questions
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31 views
Finding the image of a point by analytic continuation of regular branch of $\sqrt{π^2 + \ln^2(z)}$ in $\mathbb{C}\backslash Γ$
Denote $\varphi$ the regular branch of analytic function $\sqrt{π^2 + \ln^2(z)}$ in $\mathbb{C}\backslashΓ$ defined by $φ(1) = \pi$, and where $\Gamma$ is:
With only this information, I would like to ...
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3answers
19 views
How does this identity so elegantly combine an infinite sum in $\eta$ and an improper integral in $\Gamma$?
This is all well and good, but where did this come from?
In the article on the Gamma function, Wikipedia shows most of its alternate definitions with clear proofs, yet in the article on the Dirichlet ...
1
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0answers
38 views
Does my odd proof for the Abel sum for $\eta(-2)$ work?
EDIT: The correct answer to the Abel sum of $\eta(-2)$ has been given by the comments under this post. The focus of the question is now whether there is any sense to my method and my "proof" ...
1
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1answer
17 views
Analytic continuation of a certain family of $L$ series
Consider the function
$$
G(s,x) : = \sum_{n=1}^\infty \frac{\exp(2\pi i nx)}{n^s}
$$
where $x\in[0,1)$ and $s\in\mathbb{C}$. The series is absolutely convergent for $Re(s) > 1$.
If $x\in\mathbb{Q}\...
1
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0answers
48 views
Finding the best Taylor-esque polynomial
Background
Based only on the information provided by derivatives at a single point, the Taylor series provides the 'best'* approximation of the function around its radius of convergence. But, some ...
0
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1answer
41 views
Does the analytic continuation of $\zeta(s)$ for $s\in\mathbb{C} \ \backslash \ \{1\}$ require Cauchy's integral formula?
I've seen Cauchy's integral formula used in proofs for the analytic continuation of the Riemann zeta function to the entire complex plane ($s\neq1$). My question is whether a counterexample can be ...
4
votes
1answer
47 views
Is there an analytic continuation at $z=R$?
Assume $f(z)$ is analytic in $D_R(0)$ (in other words, the corresponding power series $\sum_{k=0}^{\infty}a_kz^k$ has the convergence radius equal to R). Suppose we make an analytic continuation to a ...
0
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1answer
54 views
Connection between $J_k= \sum_{n=1}^k e^{-n}=\frac{1-e^{-k}}{e-1} $ and $ f(x)=\sum_{n=1}^\infty e^{-n^x} ?$
Consider the geometric series$$J_k= \sum_{n=1}^k e^{-n}=\frac{1-e^{-k}}{e-1} $$
I'm wondering if this has any connection with:
$$ f(x)=\sum_{n=1}^\infty e^{-n^x}. $$
$J_k$ can be interpreted as adding ...
2
votes
1answer
52 views
Why does this factorial limit hold for complex numbers?
EDIT: it seems implied by a response to this post that a proof of this fact does actually exist, but I've yet to see the proof or any intuition for the proof - so what is it?
OP:
According to a ...
0
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1answer
32 views
Analiytic Continuation of a function
For the given curve $γ$ fix the regular in domain C\ $γ$ branch of $f(z) = ln z$ by $f(1) = 0$. For
that branch find the value of its analytic continuation f(−3) =?
I just know that we have to work ...
-2
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0answers
50 views
I'm looking for the value of the riemann zeta function at - 1/2 [closed]
Has anyone done the calculation for the value of the analytically continued riemann zeta function at zeta (-1/2)? Does it converge? Thank you, and kind regards.
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0answers
40 views
Finding analytic continuatiuon of a branch
I'm warning you that this post relates of a topic I'm not comfortable with, so in order to solve the problem I will write below, I'm very interested in understanding other cases/general cases.
The ...
2
votes
0answers
104 views
Can we interpret $\sum_{n=1}^\infty n$ to be equal to anything by analytic continuation of different functions? [duplicate]
We've all seen the claim that $\sum_{n=1}^\infty n = -1/12$, using the interpretation that $\sum_{n=1}^\infty n$ is the value of the analytic continuation of $\zeta(s) = \sum_{n=1}^\infty n^{-s}$ at $...
0
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0answers
16 views
Analytic continuation of two functions along a common domain
Suppose $f$ and $g$ are analytic in a neighborhood of $z=0$, and both admit unique analytic continuations to $z=1$, meaning no matter how you analytically continue $f$ and $g$ to a neighborhood of $z=...
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0answers
11 views
How can I make analytic continuation across the branch cut to a function?
Suppose I have a function $G(z)$ with choosen branch cut, say negative real axis, and it's form is known. Now I want to make analytic continuation to $G(z)$ from upper half complex plane to the lower ...
1
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1answer
47 views
Example on Analytic continuation
Given
$$f_1(z)=\int_0^\infty t^{z-1}e^{-t}dt$$
where $\operatorname{Re}(z)>0$
$$f_2(z)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+z)(n!)}+\int_1^\infty t^{z-1}e^{-t}dt$$
except for the values $z=0,-1,\ldots$...
0
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0answers
30 views
Analytic continuation and convergence of generating function
For a real sequence $(a_n)_{n\in\mathbb{N}}$ I am considering the generating function $s(z) = \sum_{n=0}^{\infty} a_n z^n$ and by some calculations I find that it has a representation $f:\mathbb{C}\to ...
1
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1answer
37 views
two disjoint compact sets, holomorphic function there exists a decomposition $f=f_1+f_2$
Let $D_1$ and $D_2$ be two compact sets in $\Bbb C$, $D_1\cap D_2=\emptyset$, and $ f\colon \Bbb C\setminus(D_1\cup D_2)\to\Bbb C$ be a holomorphic function. Show that there exist two holomorphic ...
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59 views
Analytic continuation of a periodic function on the real line
Suppose $f(x+1)=f(x)$ is real analytic for $x\in \mathbb{R}\setminus \mathbb{Z}$. Moreover, it is once-differentiable at $x\in \mathbb{Z}$.
In addition, suppose that on the interval $(0,1)$, we know ...
1
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0answers
163 views
Extending $\sum_{n=0}^\infty s^{n^2}$ beyond its natural boundary
Let $\mathbb{D} = \{s \in \mathbb{C} : |s| < 1\}$. Let $f : \mathbb{D} \rightarrow \mathbb{C}$ where
$$ f(s) = \sum_{n=0}^\infty s^{n^2} $$
$f$ is analytic on $\mathbb{D}$. This is what it looks ...
0
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0answers
35 views
Analytic continuation of a periodic function on the real line using a Bernoulli polynomial
Consider a function $\Gamma(z;\tau,\sigma)$ which is meromorphic in $z$ and periodic under $z\to z+1$.
I am interested in a certain limit of this function, where $\tau$ and $\sigma$ take specific ...
0
votes
1answer
37 views
Prove that an analytic function is zero
Suppose that I have an analytic function in four variables $$f = f(z_1,z_2,z_3,z_4)$$
such that I know the following facts:
$$\Re [f(z_1,z_2,0,0)] = 0$$
and that:
$$f(0,0,z_2,z_3) = 0$$
I was ...
0
votes
1answer
62 views
Closed form for sums containing exponential function:
How to get closed form of following sums:
$$\sum_n e^{-n/2}n^{k-1}\left(s-\frac{1}{mn}\right)^k$$
$$\sum_n \frac{e^{-n/2}}{n^2\left(s-\frac{1}{mn}\right)}$$
Here $s,k,m$ are constants and $n$ runs ...
1
vote
1answer
47 views
computing $\int_{-\infty}^{\infty} x e^{itx} dx$ using Cauchy Theorem and gamma function rapresentation
I want to solve the above integral with $t \in \mathbb{R}$; using a software (Wolfram) to verify the result I obtain the following result:
$I(t) \equiv \int_{-\infty}^{\infty} x e^{itx} dx = -i \frac{...
1
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0answers
63 views
Analytic continuation of $p(s)=\sum_{p\text{ prime}} e^{-p^s}$
What is the analytic continuation of $p(s)=\sum_{p\text{ prime}} e^{-p^s}=e^{-2^s}+e^{-3^s}+e^{-5^s}+\cdots$ ?
I've attempted the case where the index of the sum is not prime, but is $n=1,2,3,\dots$. ...
1
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0answers
58 views
On evaluating $\rho(s)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma(z)P(sz)\,dz.$
How do you evaluate the following integral?
$$\rho(s)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma(z)P(sz)\,dz.$$
$P(\cdot)$ is the prime zeta function. I got the integral from taking $\rho(s)=...
0
votes
0answers
27 views
Clarification on analytic continuation of polylogarithm definition
I am trying to understand the branching geometry of the Dilogarithm function. In the Google book:Functional equation of hyperlogarithms, equation 8.6 written for the special case $n=2$ gives:
$$
\...
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0answers
64 views
Where is $\varphi(s)=\Gamma\left(1+\frac{1}{s}\right)+\sum_{n\ge0} \frac{(-1)^n}{n!}\zeta(-ns)$ equal to zero?
It's known that when you analytically continue the Riemann zeta function to the complex plane, it has roots.
I would like to figure out if there are roots to the following analytic continuation of $\...
0
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0answers
9 views
Calculate the best time to add interest to the invested capital
I have a game that goes like this:
I have a starting bet of, say, \$1,000.
I have a 30% APR interest which is on a separate account.
I can withdraw my interest at any time and add it to the principal ...
1
vote
1answer
71 views
build complex manifold by analytically continuing set of real analytic functions on a real analytic manifold?
Take a real analytic surface embedded in $\Bbb R^3$, homeomorphic to a sphere, and analytically continue each real analytic function (the union of which equals the surface) to the complex space $\Bbb ...
2
votes
1answer
22 views
Extension of real bounded analytic function to a holomorphic bounded function.
Let $f(x)$ be a real-valued analytic bounded function. Can I find a open set $U\subset\mathbb{C}$, containing $\mathbb{R}$, and a holomorphic complex-valued function $\bar{f}(z)$ such that $\bar{f}(z)$...
1
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1answer
38 views
Is there a continuous, or smooth, generalization of the iterated logarithm $\log^* n$?
I was wondering about the properties of certain classes of functions related to algorithmic runtimes in this post.
So I know that $n!$ has a continuous/smooth generalization to the gamma function. Is ...
1
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0answers
24 views
understanding analytic continuation along a path using germ
Let $\mathbb{O}$ is the sheaf of germs of holomorphic functions on $\mathbb{C}$, $p: \mathbb{O} \rightarrow \mathbb{C}$ defined by sending the germ $f_a$ to $a \in \mathbb{C}$.
Given a path $\gamma: [...
2
votes
0answers
34 views
Analytic Continuation of Fractions
Excuse my lack of expertise, I study natural sciences (physics) and not mathematics so I will be off with my terminology and mathematical vocabulary. Please feel free to poke and build at this idea ...
0
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0answers
44 views
Explicitly writing out the analytic continuation of a real valued function
Consider the following function
$$
f(x) = x\cdot h(x) - g(h(x))
$$
where $x\in\mathbb{R}, f:\mathbb{R} \to \mathbb{R}^{+}$ and $f, h, g$ are analytic everywhere and extend to be complex analytic on $\...
2
votes
2answers
94 views
What does meromorphic continuation mean?
Sorry if this question is slightly out of context.
I've learned that $\zeta(k)=\sum\limits^{\infty}_{z=1} \frac{1}{z^k}$ has meromorphic continuation to $Re(s)>0$.
I know that, from convergence ...
1
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1answer
79 views
Hartog's extension theorem for codimension 2
I need to use a version of Hartog's extension theorem, that I did not find by googling around. However I think I found a solution myself, and wanted to ask if this is correct. Any feedback would be ...
0
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0answers
45 views
Where does $s-1$ come from in this $\zeta(s)$ equation?
I have been working my way through this Arxiv paper concerning the analytic continuation of the zeta function. I don't understand the first equality in equation (19), page 6.
In equation (11), the ...
0
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1answer
24 views
Determine the largest domain to which this function can be analytically continued
Consider the following question which was asked in my Complex analysis Quiz( which is now over) :
Consider the following analytic function $f(z)= \sum_{n=1}^{\infty}\frac{1+c}{n+c} z^n $ Determine ...
2
votes
1answer
76 views
Analytic continuation of a series involving $\zeta(s)$?
Context:
In this this paper, Juan Arias de Reyna introduces on p12 the following equation for a new power series coefficient:
$$A_m = \lambda_m + \frac{\gamma + \log(4\pi)}{2}-\frac{1}{m}-\frac{1}{m}\...
0
votes
2answers
40 views
Divergence of $\zeta(z)$ tamed or not tamed by any analytic continuation
We know the conjecture about the Riemann hypothesis is about the nontrivial zeros are on
$$(1/2 + r i)$$
for some $r \in \mathbb{R}$ of Riemann zeta function.
My question is how many divergences of $\...
2
votes
1answer
83 views
Convergence of $\int_0^{\infty} s^{2-2\alpha}e^{is^2}ds$
I need to evaluate the following integral
$$\int_0^{\infty} s^{2-2\alpha}e^{is^2}ds$$
where $\alpha\in[0,\,1]$. I shall call this integral Gamma-function like integrals since it can be brought to ...
3
votes
2answers
40 views
Does the branch of the initial value determine the branch for the whole curve?
Let $\Omega$ be a connected and simply-connected open subset of $\mathbf{C}$ which does not contain the origin. Let $\gamma :[0,1] \to \Omega$ be a simple smooth curve and let $\zeta := \gamma(0)$. ...
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0answers
90 views
Assigning other finite values to $1 + 2 + 3 + …$
Can finite values other than $-1/12$ be assigned to the divergent sum of all natural numbers in a similar way? To define precisely what would be the "similar way", consider the following:
...
5
votes
0answers
175 views
Analytic continuation of the Kronecker Delta
The Kronecker Delta can be written as the integral of the complex function
$$f(n,z)=\frac{1}{2\pi i} z^{n-1} \ ,$$
where $n\in \mathbb{Z}$ and $z\in\mathbb{C}$ on a closed path $\mathcal{C}$ enclosing ...
0
votes
0answers
28 views
Analytic continuation of Convergent integral
I was trying to solve the following integral.
$$I = \oint _{|z|=1}\frac{dz}{2 \pi i z}\int_{0}^{\infty} dr \dfrac{e^{-\tfrac{r^2}{z^2}}r^{2n+1}}{z^2(z-1)} $$
The singular structure in the $z$ ...
1
vote
1answer
125 views
Analytic continuation of square root with branch cut along the negative imaginary axis by using a suitable logarithmic branch.
We are working with real integrals and the complex residue theorem. In order to solve the following integral:
$$ \int_0 ^\infty \frac{\sqrt{x}}{1+x^2},$$
I will have to think about how a square root ...
0
votes
0answers
20 views
How to prove that set of regular points of an analytic function is open
This question is from Pg 457 of Ponnusamy and Silvermann's Complex analyis book.
How to prove that set of regular points of an analytic function is open ?
$z_1$ is called regular point wrt analytic ...
1
vote
1answer
36 views
What does monodromy theorem tells about log z and $z^{1/2}$
This question was asked in my complex analysis assignment and I am confused on how to attempt it.
What does monodromy theorem tells about log z and $z^{1/2}$?
I have to choose a Domain D and then ...
2
votes
0answers
28 views
Can infinitely many points on the boundary C of a domain be singular without C being a natural boundary
This question was asked in my complex analysis quiz and I was unable to do it.
Can infinitely many points on the boundary C of a domain be singular without C being a natural boundary ?
I thnik it can ...
