Questions tagged [analytic-continuation]

For questions related to analytic continuation

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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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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 ...
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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" ...
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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}\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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 ...
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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 $...
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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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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 ...
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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 $z=0,-1,\ldots$...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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{...
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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 $n=1,2,3,\dots$. ...
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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)=...
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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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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 $\...
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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 ...
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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 ...
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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 $\bar{f}(z)$...
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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 ...
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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: [...
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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 ...
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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 $\...
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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 ...
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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 ...
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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 ...
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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 ...
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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}\...
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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 $\...
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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 ...
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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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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: ...
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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 ...
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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 $z$ ...
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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 ...
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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 ...
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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 ...
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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 ...

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