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Questions tagged [analytic-continuation]

For questions related to analytic continuation

11
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2answers
938 views

On every simply connected domain, there exists a holomorphic function with no analytic continuation.

I am working on a question that requires me to prove that on every simply connected open subset of $\mathbb{C}$, there exists a holomorphic function that cannot be extended to a holomorphic function ...
10
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7answers
3k views

Intuition behind $\zeta(-1)$ = $\frac{-1}{12}$ [duplicate]

When I first watched numberphile's 1+2+3+... = $\frac{-1}{12}$ I thought the sum actually equalled $\frac{-1}{12}$ without really understanding it. Recently I read some wolframalpha pages and watched ...
9
votes
1answer
204 views

Exercise 2 from Terry Tao's blog on Euler-Maclaurin, Bernouilli numbers, and the zeta function

In the blog post The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation, Terry Tao looks at the commonly-cranked 'absurd' formulae $$\begin{align} \...
9
votes
1answer
160 views

Expressing Zeta function using Gamma series

Motivated by Gautschi double inequality, $$ \frac{n^{s}}{n^{\small1}}\ge\frac{\Gamma(n+s)}{\Gamma(n+1)}\ge\frac{(n+1)^{s}}{(n+1)^{\small1}}\ge\frac{\Gamma(n+1+s)}{\Gamma(n+1+1)}\ge\,\cdots \quad\...
8
votes
1answer
333 views

Fourier transform of meromorphic function

Suppose that I have a function $f(z)$ which is meromorphic on the entire complex plane, meaning holomorphic everywhere except for a discrete set of poles. I then take a vertical slice of this ...
8
votes
1answer
165 views

Does there necessarily exist such a holomorphic function?

This is an old qual problem I'm working on: Let $f:[0,1]\rightarrow \mathbb{R}$ be a $C^{\infty}$ function. Does there necessarily exist a holomorphic function $g: \mathbb{C}\setminus\{0\}\rightarrow \...
7
votes
7answers
293 views

If $|f| \le |g|$, does analytic continuation of $g$ imply analytic continuation of $f$?

Let $f,g$ be two holomorphic functions on a domain $D$ such that $|f(z)| \le |g(z)|$ for all $z \in D$. Further suppose that there is an analytic continuation of $g$ to a bigger domain $D'$. Does that ...
7
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3answers
412 views

Can we use analytic continuation to obtain $\sum_{n=1}^\infty n = b, b\neq -\frac{1}{12}$

Intuitive question It is a popular math fact that the sum definition of the Riemann zeta function: $$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} $$ can be extended to the whole complex plane (except ...
7
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1answer
141 views

On the complex function $f(s)=\sum\limits_{n=1}^\infty\sigma(n)^{-s}$

Let $s=x+iy$ the complex variable (if you want a difffernt notation you are welcome), then I know that $\sum_{n=1}^\infty n^{-s}$ converges for $\Re s>1$. On the other hand let $\sigma(n)=\sum_{d\...
6
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2answers
365 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. ...
6
votes
1answer
529 views

Is there a holomorphic function $f$ on the unit disc such that $|f(z)|\rightarrow\infty$ as $|z|\rightarrow 1$?

When I learnt that there exists a holomorphic function on the unit disc $D$ that cannot be continuously extended to a domain that is strictly larger $D$, I was taught the example $$z\mapsto\sum_{n=1}^\...
6
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1answer
189 views

Riemann zeta-function regularization in string theory

First of all let me say that I am a physicist and therefore it is sometimes hard for me to understand some mathematical steps... Now, I've been trying to obtain the well known result for the zeta ...
6
votes
0answers
316 views

Why do the Jacobi theta functions have a natural boundary?

The Jacobi theta functions, like $$ \theta_3(z,q)=1+2\sum_{n=0}^\infty q^{n^2}\!\cos(2nz) , $$ look relatively innocent in how they handle the 'nome' $q$, a complex parameter that shapes the ...
6
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0answers
610 views

Show that $f(z)=\sum_{n=0}^{\infty}z^{2^n}$ can't be analytically continued past the unit disk.

I'm reading the problems of Stein and Shakarchi's Complex Analysis, Chapter 2 Problem 1 asks to show that $$f(z)=\sum_{n=0}^{\infty}z^{2^n}$$ cannot be analytically continued past the unit disk. (...
5
votes
1answer
78 views

How is extension by analytic continuation done?

I understand that by defining an expression for a function to be analytic, we can extend the range of the expression beyond it's usual range. One case is that a series which is asymptotic to a ...
5
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1answer
166 views

Prove that the given condition implies analytic continuation

Here is an old qual problem I'm working on, I have some idea, but I'm not sure if I'm correct or not. I would be happy if anyone could possibly confirm or correct me: Let $U=\{z\in \mathbb{C} : \frac{...
5
votes
1answer
101 views

Let $f(z)$ be a function analytic in a domain containing the segment $[0,1]$ and satisfying $f(z+1)=azf(z)+p(z)$.

Let $f(z)$ be a function analytic in a domain containing the segment $[0,1]$ and satisfying $$ f(z+1)=azf(z)+p(z) $$ in that domain, where $a\in\mathbb{R}$ and $p$ is a polynomial. Show that $f$ ...
5
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0answers
121 views

Is $\int_0^\infty \frac{dt}{e^t-xt}$ analytic continuation of $\sum_{k=1}^\infty \frac{(k-1)!}{k^k} x^{k-1}$?

The following power series apparently converges only for $-e \leq x <e$: $$f(x)=\sum_{k=1}^\infty \frac{(k-1)!}{k^k} x^{k-1}$$ We can use it to define a real function $f(x)$, analytic in that ...
4
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2answers
674 views

Alternating series test for complex series

I want to show that we can continue Riemann's zeta function to Re$(s)>0$, $s\neq 1$ by the following formula \begin{align} (1-2^{1-s})\zeta(s)&=\left(1-2\frac{1}{2^s}\right)\left(\frac1{1^s}+\...
4
votes
2answers
281 views

Alternative analytic continuation to zeta, not giving $-\frac{1}{12}$ for sum of integers

Apologies if this has been asked already. Inspired partly by this answer where an $n e^{-\epsilon n}$ rather than $n^s$ regularization was made in the 'evaluation' of $\sum\limits_{n=1}^{\infty}n$ and ...
4
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2answers
223 views

Analytic continuation for $\zeta(s)$ using finite sums?

$\zeta(s)$ converges for $\sigma >1$ but not for $\sigma =1/2.$ But for some reason for $s = 1/2 + i t $ and fixed finite $N,~$ $\zeta_N(s) =\sum_{n=1}^N\frac{1}{n^s}$ is very close to $\zeta(s)$ ...
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|> ...
4
votes
2answers
129 views

Evaluate the integral $\int_0^\infty x^{t-1}e^{-\beta x}dx$

I want to evaluate the following integral $$\int_0^\infty x^{t-1}e^{-\beta x}dx$$ where $\beta$ is a complex number. Now, if $\beta$ was real, we could just set $y = \beta x$ and we will reduce to ...
4
votes
2answers
83 views

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 ...
4
votes
1answer
100 views

Analytic Continuation for the case : $\sum_{1}^{\infty} 1$

Analytic Continuation for the case : $\sum_{k=1}^{\infty} 1$ INTRODUCTION $\ \ \ \ \ $Find a convergence on the sum $1+1+1+1+...$ through analytic continuation of the series as a special case of $1+x+...
4
votes
1answer
58 views

Analytic extension of $\sum_{k=1}^n\frac1k$ complex domain

The analytic extension: $$\sum_{k=1}^n\frac1k=\int_0^1\frac{x^n-1}{x-1}dx$$ I was wondering for what values of $n$ does this extend to, mainly complex values of $n$. I know it is defined for $n=0$, ...
4
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1answer
280 views

Does a branch cut discontinuity determine a function near the branch point?

Suppose $g(z)$ is analytic on a disc centered at the origin, except along the negative real axis where it has a branch cut discontinuity. Also assume that $g(0)=0$. Let $h(x)$ give the discontinuity ...
4
votes
3answers
493 views

Gamma & Zeta Summation $\sum_{n=0}^{\infty}\frac{\Gamma(n+s)\zeta(n+s)}{(n+1)!}=0$

According to Gamma Summation & Zeta Summation: $$ \sum_{n=0}^{\infty} {(-1)^n \frac{\Gamma(n+s) \zeta(n+s)}{(n+1)!}}=\Gamma(s-1) \quad : \space Re\{s\}<2 $$ Show that: $$ \sum_{n=0}^{\...
4
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0answers
31 views

Translation of integration contour

I have an argument where, with a meromorphic function $\phi$ satisfying $\phi(s) = \phi(1-s)$, the following equality appears: $$\int_{(2)} \phi(s) y^{-s} \, ds = \int_{(1/2)} \phi(s) y^{-s} \, ds$$ ...
4
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0answers
87 views

Is there a natural topology in which the analytic continuation of a series converges everywhere it is defined?

The expression $$ \sum_{n=1}^\infty n = -\frac{1}{12} $$ has been met with considerable controversy. Previous questions (such as this and this) searching for a way of rationalizing the formal ...
4
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0answers
141 views

Black magic behind Padé approximation

In condensed matter physics one usually calculates the so-called Matsubara Green's function in the set of discrete points $G (i \omega_n)$, where $$ \omega_n = (2n+1) \pi T $$ Physically significant ...
4
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0answers
44 views

Analyitc Continuation of Partial $\zeta$ function

Let $A\subset \mathbb{N}$. Consider the partial $\zeta$ function $$\sum_{a\in A} \frac{1}{a^s}$$ We know this converges for Re$(s)>1$. Under what conditions of $A$, can this be analyically ...
3
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1answer
40 views

Is there an analytically continued function of $z^p$ at zero?

For a rational number $p > 1$. We know that the function $z^p$ is holomorphic on $\mathbb{C} \setminus \mathbb{R}^-$ (excluding $z = 0$). Is there an analytic continuation of the function $z^p$ at ...
3
votes
2answers
521 views

How can I know the analytic continuation exists in certain cases?

As pointed in Does the analytic continuation always exists? we know it doesn't always exist. But: take the $\Gamma$ function: the first definition everyone meet is the integral one: $$ z\mapsto\int_{...
3
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1answer
141 views

On Alexander polynomial of a knot

The Alexander polynomial of a knot is of the form $$\Delta(t)=det(V^T-tV),$$ where $V$ is the Seifert matrix, see http://archive.lib.msu.edu/crcmath/math/math/a/a116.htm. What is geometric or some ...
3
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2answers
1k views

Calculating values of the Riemann Zeta Function

The Riemann Zeta Function is most commonly defined as $$\zeta(s)=\sum_{n=0}^\infty \frac{1}{n^s}$$ There is some sort of million dollar prize that involves proving the real part of complex number s ...
3
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1answer
79 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}{...
3
votes
2answers
129 views

Analytic continuation of $\sum (z/a)^n$

I'm having trouble continuing this function beyond its convergence radius, $R=a$. $$f(z)=\sum (z/a)^n$$ Given the context (a textbook in complex analysis) I suspect it should have a simple closed-...
3
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1answer
94 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 ...
3
votes
1answer
73 views

Extension of a conformal mapping to an elliptic function

Let $\rho$ be the conformal mapping from the interior of the triangle with vertices $-1,i\sqrt{3},1$ onto the upper half-plane. Show that $\rho$ has an elliptic extension. The hypothesis makes sense ...
3
votes
1answer
48 views

Analytic Continuation for a Product

I was trying to solve the functional equation $$\phi(x)^2\phi(2x)=x^2+2x+1$$ and by assuming that $\phi(1)=1$, and setting up a recurrence relation, I found the solution $$\phi(x)=\prod_{i=0}^{\log_2(...
3
votes
1answer
601 views

Analytical continuation of complete elliptic integral of the first kind

I am dealing with a problem involving the complete elliptical function of the first kind, which is defined as: $K(k)=\int_0^{\pi/2} d\theta \frac{1}{\sqrt{1-k^2\sin^2(\theta)}}=\int_0^1 dt \frac{1}{\...
3
votes
1answer
207 views

Analytic continuation of $\sum z^{2n}$

If I'm not misusing the root test, the convergence radius of $\sum z^{2n}$ is $\lim \sup \sqrt[2n]{1}=1$ (is this correct?). Now, is there a closed-form expression of $f(z)=\sum z^{2n}$ so that it be ...
3
votes
1answer
109 views

Analytic continuation of power series on the unit whose terms tends to 0

This problem is from complex analysis. Set $$f(z)=\sum_{n=0}^{\infty}a_nz^n$$ with convergence radius of 1, and $$\lim_{n \to \infty}a_n=0$$ Prove that if $z_0 \in \partial B(0,1)$ is not a singular ...
3
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1answer
271 views

Value of polylogarithms $\mathrm{Li}_s(1)$ for $s<1$.

For real $s>1$ the polylogarithm for $z=1$ reduces to the Riemann zeta function $\mathrm{Li}_s(1) = \zeta(s)$. For real $s<1$ Wolfram Alpha and the Wolfram function site give $\mathrm{Li}_s(1) = ...
3
votes
3answers
404 views

Numerically solve complex differential equation

I'd like to find a numerical solution to a complex differential equation of the form $ \frac{dz(t)}{dt} = f(z,t)$, where $z$ and $t$ can both be complex, with $z(0)=0$. Specifically, I'd like to ...
3
votes
1answer
94 views

Stability result for analytic continuations

Let $f(x):\mathbf{R} \rightarrow \mathbf{R}$ be a real function which extends meromorphically to the complex $\mathbf{C}$ plane to a function $\tilde f(z) : \mathbf{C} \rightarrow \mathbf{C}$. Let ...
3
votes
2answers
55 views

Analytic continuation commuting with series

Suppose $f_1,f_2,...$ are entire functions, and there is an open subset $U \subseteq \mathbb{C}$ such that the series $F(z) = \sum_{n=1}^{\infty} f_n(z)$ converges normally on $U$. Also suppose that $...
3
votes
0answers
21 views

Meromorphic continuation of the multifactorial

The double factorial $z!!$, like the normal factorial function, can be extended to the complex plane using $$ z!! = 2^{z/2} \left(\frac{\pi}{2}\right)^{(\cos\pi z-1)/4} \left(\frac{z}{2}\right)! $$ ...
3
votes
0answers
40 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 ...