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Questions tagged [analytic-continuation]

For questions related to analytic continuation

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Are all complex algebraic varieties in 2+ variables unbounded as a result of Hartogs' extension theorem?

Hartogs' extension theorem states that for any $n\geq 2$, $U\subseteq\mathbb C^n$, $K\subset U$ compact (in $\mathbb C^n$) and a holomorphic function $f:U\setminus K\to\mathbb C$, it can always be ...
Boris Dimitrov's user avatar
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Is there some link between analytic continuation and potential theory?

It seems like the analytic continuation of a function has a lot in common with the process of trying to define a potential for a vector field (or a differential 1-form). In particular, an analytic ...
Leonardo Rossi's user avatar
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Finding the natural boundary of $\sum_{n=0}^{\infty}\frac{z^n}{n^k}$ for $k \ge 2$

Let $k\ge 2$, given $$f(z)=\sum_{n=1}^{\infty}\frac{z^n}{n^k}$$ It is easy to see that it converges for $|z|\le 1$, but how can it be analytically continued beyond the unit circle? Hadamard proved ...
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Proposition 16.5.4 in Ireland-Rosen

We aim to show that if $\chi$ is a complex Dirichlet character mod $m$, then $L(1, \chi) \neq 0$. Assuming otherwise, we easily prove that if $F(s) = \prod_{\chi}L(s, \chi)$, where the product is over ...
Johnny Apple's user avatar
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Problem 6.2.10 From Complex Analysis by Jihuai Shi

Definition: let $f$ be holomorphic on a region $G$ (here region means a non-empty connected open set). For $\xi\in \partial G$, if there exists a ball $B(\xi,\delta)$ and a holomorphic function $g \in ...
Robert's user avatar
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Analytic continuation for $L$-series with real character in Murty

In Problems in Algebraic Number Theory from M. Ram Murty, he mentions on page 148 how one show non-vanishing at 1 for a certain $L$-series. Here, he works with a generalization of Dirichlet $L$-series ...
Logwan27's user avatar
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Literature check for a summation formula

In this paper I did a lot of stuff relating to sums, including a formula for explicitly evaluating them near the end of the paper: Let $f(x)=\sum_{k=0}^\infty c_kx^k$ be the power series of $f(x)$. ...
Kamal Saleh's user avatar
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maximal continuation of $\Pi_2(x)$

Consider the functions for $k\in \Bbb N$ $$ \Pi_k(x) := \sum_{n \in \Bbb N} e^{\frac{\log^k n}{\log x}} $$ $\Pi_1(x)$ converges for real $1/e<x<1$. $\Pi_1(x)$ is a Riemann zeta function i.e. $\...
zeta space's user avatar
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solution verification: Is $K(s)$ holomorphic on $\Bbb C$?

Consider the Mellin integral $$K(s)=\int_{1/2}^1 \zeta\bigg(-\frac{1}{\log x}\bigg)~x^{s-1}~dx $$ Where $\zeta(\cdot)$ is the Riemann zeta function defined for real $1/e<x<1.$ $K(s)$ is ...
zeta space's user avatar
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Proof of analytic continuation on manifold

I was reading the theorem below in this article. My question concerns the passages in bold. Specifically, why would Z be open? Is it because we can use germs to define a topology where Z would be the ...
yumika's user avatar
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Meromorphic continuation of Euler product

Short version: What can be said about the meromorphic continuation of the Euler product $$\prod _{p}\left (1+\frac {p^{-s}}{p-2}\right )?$$ Longer version: I realise I have some misconceptions about ...
tomos's user avatar
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analytically continue functions on independent planes in $\Bbb R^3$

I know techniques to analytically continue some functions off the positive real line. I am less sure given I have 2 independent euclidean planes sitting in $\Bbb R^3$ with some function $f$ defined on ...
zeta space's user avatar
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Funny lattice equivalence between additive and subtractive extended square roots

I am interested in the behavior of the "extended square root", such as: $$\sqrt{2+\sqrt{2+\sqrt{2+\sqrt2}}} ...$$ $$\sqrt{n\pm\sqrt{n\pm\sqrt{n\pm\sqrt{n\pm}}}} ...$$ I've noticed after ...
Ivy Darling's user avatar
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How to find the divergent renormalization of $\sum_{n=1}^{\infty} \frac{2^n}{n}$?

Consider the divergent series $$\sum_{n=1}^{\infty} \frac{2^n}{n} $$ This can be seen as arising from the function $f(z) = -\ln(1-z)=\sum_{n=1}^{\infty} \frac{z^n}{n} $ and 'evaluating' that power ...
Sidharth Ghoshal's user avatar
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How to evaluate $\int_{-\infty}^\infty \sin^3(x)/x^3 dx$ without finding an analytic continuation, still using complex analysis.

There is a similar post regarding the integral $\int_{-\infty}^\infty \sin^3(x)/x^3 dx$ on Stack. The reason I have some trouble with this post is because it gives an analytic continuation of the ...
Luk'yan Vilshansky's user avatar
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288 views

Analytical continuation of a Matsubara sum

I want to numerically calculate the low temperature limit of the following Matsubara sum $$ S(\Omega) = \pi T \sum_{\omega_n} \frac{4\Delta^2+\Omega^2}{s_1 s_2 (s_1 + s_2)},$$ with $\omega_n = \pi T (...
Jean's user avatar
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Do analytic continuations exist over $\mathbb{R}$?

I have a feeling this is quite obvious since $\mathbb{R}\subset\mathbb{C}$, but suppose we have the function $f:\left[-1,1\right]\times\mathbb{R}\to\mathbb{R}$ defined by $$f\left(x,y\right) = \frac{y\...
gettingmathy's user avatar
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Proof of ellipticity of lemniscate functions from integral definition

The lemniscate functions $\text{sl}$ and $\text{cl}$ are the solutions to the differential equation $$ (y')^2+y^4=1$$ with $\ y(0)=0, \ y'(0)=1$ $\ $ or $\ $ $y(0)=1, \ y'(0)=0.$ Using the integral ...
Noa Arvidsson's user avatar
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Analytic continuation of the cardinality of a set: -1/2 element sets? Sets with an imaginary number of elements?

The number of r-element subsets of a set of cardinality n is given by $_nC_r$, with is equivalent to the factorial expression $$\frac{n!}{r!(n-r)!}$$ Well, whenever I see a factorial I get the urge to ...
Alexandra's user avatar
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Analytic continuation of $(x-a)^{-1/4}$?

In Landau's book on quantum mechanics an expression of the form $$ (x-a)^{-1/4} $$ is analytically continued around the upper half-plane along the semicircle from $0$ to $\pi$. The result stated is $$(...
hhh3's user avatar
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Analytically continuing a function of two complex variables.

I am aware of the identity theorem and how it allows us to extend the definition of a complex analytic function from $A \subset \mathbb{C}$ to a larger domain $D$ that is an open subset of the complex ...
mathphy24's user avatar
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Analytic continuation of Bernoulli numbers: $B_{1/2}$?

$B_0$ is 1, $B_1$ is $\pm 1/2$, and so on (I'm not going to list every single Bernoulli number). The Bernoulli numbers $B_n$ are defined for every integer $n \ge 0$. But what about $B_{1/2}$? And $B_i$...
Alexandra's user avatar
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Struggling to Understand Analytic Continuation Lemma

I'm confused about a Lemma in Ian Stewart & David Tall's Complex Analysis: Lemma 14.1: Let $f , g$ be analytic on a domain $D$. Suppose that $P$ and $Q$ are open sets and $∅\ne P ∩ Q ⊆ D$. ...
Sam's user avatar
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A question about the complex function's continuity

Suppose that $D$ is a connected open set, $n$ is a natural number. Show that: $f\in C^{n}$ if and only if $\frac{\partial^n f}{\partial z^k\partial\bar{z}^{n-k}}(0\leq k\leq n)$ is continuous on $D$. $...
Er Bu's user avatar
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Showing that $f\in C^{n}$ if and only if $\frac{\partial^n f}{\partial z^k\partial\bar{z}^{n-k}}$ is continuous

Suppose that $D$ is an area, $n$ is a natural number. Show that: $f\in C^{n}$ if and only if $\frac{\partial^n f}{\partial z^k\partial\bar{z}^{n-k}}$ is continuous on $D$. It's easy to prove ...
Er Bu's user avatar
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Analytic continuation of function given by Moser-de Bruijn sequence

I was wondering about the function $$F(x) = \prod_{n=0}^{\infty}{(1+x^{4^{n}})} = 1+x+x^4+x^5+x^{16}+x^{17}+...$$ where the exponents in the resulting power series are given by the Moser-de Bruijn ...
HeliumHydride's user avatar
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How to interlopate the pickover factorial?

Question I want to be able to interlopate the pickover factorial, defined as: $$n$ = ^{n!}n!$$ where $^xx$ represents tetration. I want to be able to interlopate this function. Context The reason I ...
SebbyIsSwag's user avatar
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Is the hypersurface of revolution associated to the generator $\zeta_d(t)$ related to the primes/Riemann zeta function?

Let $Y:=[0,1]$ and $X:=[0,t]$ and consider the following curves embedded in $Y^{d+1}:$ $$ \zeta_1( t)=\bigg(\lim_{k \to\infty}\bigg( \int_{Y}\sum_{n=1}^k\varphi_n(x)dx\bigg)^{-1}\int_{X} \sum_{n=1}^k \...
zeta space's user avatar
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Unique continuation for first order equation IVP

Let $\alpha \in (0,1)$ and $\eta \in C^{1,\alpha}(-1,1)$ be a solution to $$ \eta'(x) = w(x,\eta(x)),\qquad \eta(x)=0 \quad\mbox{for }x \in (-1,0],\qquad \eta'(x)=0 \quad\mbox{for }x\in (-1,0], $$ ...
Gio712's user avatar
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Analytical continuation of $\zeta(s)$ at $\operatorname{Re}(s) = 0$

$\underline{\smash{\textbf{Background:}}}$ It is a well-known fact that the original definition of the zeta function, namely $$ \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} $$ is only defined when $\...
soggycornflakes's user avatar
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Analytically continuing a function defined via an integral

Suppose we want to analytically extend a function $g(z)$ that is defined as an analytic function everywhere in the complex $z$ plane except the negative $z$ axis (for example, via an integral). Would ...
mp62442's user avatar
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About the Lacunary $f(z)$ such that $f(z) = b^{-1}(a(z)), a(z) = \sum_{0<n} \frac{x^{n^2}}{(4n)!} , b(z) = \sum_{0<n} \frac{x^{n^2}}{(4n+3)!}$?

Let $$a(z) = \sum_{0<n} \frac{x^{n^2}}{(4n)!}$$ $$b(z) = \sum_{0<n} \frac{x^{n^2}}{(4n+3)!}$$ Both have a natural boundary at $|z| = 1$. Let $c(z)$ be the series reversion of $b(z)$. So $c(z)$ ...
mick's user avatar
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Analytic continuation possible here?

Consider a function $f(z)$ defined for all complex $z$ within the open unit circle. More precisely $$f(z) = a_0 + a_1 z + a_2 z^2 + \cdots$$ where the $a_n$ are complex numbers. So $f(z)$ is a Taylor ...
mick's user avatar
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What does it mean for a distribution to be analytic and how do we analytically continue it?

Let $T \in S'(\mathbb{R}^n$) be a tempered distribution. I've seen a few resources (such as the first sentence of this Wikipedia article) refer to distributions being analytic or how one can ...
CBBAM's user avatar
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When is $f(z)$ analytic in the neigbourhood of $z$ for some complex $z$ close to the real line?

Let $f(x)$ be real-valued, 1-periodic, bounded and $C^{\infty}$ for real $x$. When is $f(z)$ analytic in the neigbourhood of $z$ for some complex $z$ close to the real line ? Notice this is similar to ...
mick's user avatar
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4 votes
3 answers
149 views

How to define this function so that it is continuous?

I have a function $$f(x)=\frac{\arctan(2 \tan (x))}{x}$$ or in general for some $c\in \mathbb{R}$ $$f(x)=\frac{\arctan(c \tan (x))}{x}$$ Is there a way to define it so that it is continuous on ...
azerbajdzan's user avatar
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For all complex $|z| \neq 1$ : $\frac{z}{1+z+z^{2}+z^{3}+z^{4}} = \sum_{n=0}^{\infty} T_n \frac{z^n}{1+z^n+z^{2n}}$?

Inspired by this one For all complex $|z| \neq 1$ : $\sum_{n=0}^{\infty} w_n \frac{z^n}{1+z^n+z^{2n}+z^{3n}+z^{4n}} = \sum_{n=0}^{\infty} u_n \frac{z^n}{1+z^n+z^{2n}}$? It made sense to me, to take ...
mick's user avatar
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$ f(s) = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{p_n}= 0$?

Let $p_n$ be the $n$ th prime number. Let $f(s)$ be a Dirichlet series defined on the complex plane as : $$f(s) = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{p_n}= 1 + \frac{2^{-s}}{2}+ \frac{3^{-s}}{3} + \...
mick's user avatar
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7 votes
2 answers
292 views

Analytic extension of $\text{Li}_0^{(1,0)}(z):=-\displaystyle\sum_{n=1}^{\infty}\ln(n)z^{n}$ for $|z|>1$

I would like to extend the domain of the following function: $$\text{Li}_0^{(1,0)}(z):=-\sum_{n=1}^{\infty}\ln(n)z^{n}\qquad\text{where }|z|<1$$ The part in red is the series, the part in green is ...
Math Attack's user avatar
3 votes
1 answer
69 views

Extending a holomorphic function on $\mathbb{C}\backslash K$ to an entire function

Consider a compact set $K\subset \mathbb{R}$ with positive measure (i.e. $\mu(K)>0$), and for $z\in\mathbb{C}\backslash K$, define the holomorphic function $f$ on $\mathbb{C}\backslash K$ by \begin{...
IIIsomorphiii's user avatar
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48 views

For all complex $|z| \neq 1$ : $\sum_{n=0}^{\infty} w_n \frac{z^n}{1+z^n+z^{2n}+z^{3n}+z^{4n}} = \sum_{n=0}^{\infty} u_n \frac{z^n}{1+z^n+z^{2n}}$?

Ok I am a bit confused. So here comes a question, Consider a maclaurin series for $f(z)$ $$f(z) = \sum_{n=0}^{\infty} f_n z^n$$ where $f(z)$ has a radius of exactly $1$. $f(z)$ may or may not have a ...
mick's user avatar
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5 votes
0 answers
707 views

Description of Riemann surface of polynomial inverse

My question is about page 4 of the pdf of the following paper , one does $\textbf{not}$ need to read pages 1-3 of the paper to understand my question ( the $\textbf{only}$ part that needs to be read ...
porridgemathematics's user avatar
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Riemann zeta conjugate extension to real plane < 1

I found this function: $$ \zeta(\alpha+i\beta)*\zeta(\alpha-i\beta)=\prod \frac{1}{1-\frac{2cos(\beta ln{p})}{p^\alpha} + \frac{1}{p^{2\alpha}}}=\Re(\zeta(\alpha+i\beta))^2 +\Im(\zeta(\alpha+i\beta))^...
Goodie123's user avatar
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4 votes
1 answer
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Understanding Vladimir Maz'ya's Problem 72 what is $|dy|$?

In the article "Seventy Five (Thousand) Unsolved Problems in Analysis and Partial Differential Equations" by Vladimir Maz’ya Problem 72 on page 36 is given as follows: Let $C$ be the unit ...
Sidharth Ghoshal's user avatar
1 vote
0 answers
64 views

Simple example of analytic continuation along a path

I'm trying to understand the following example of analytic continuation given in a physics paper. In the last paragraph of page 16 the authors define the function $$f(z,\bar z)=(z\bar z)^{-\Delta}$$ ...
user avatar
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Riemann continuation theorem

Let $f\rightarrow \mathbb{C} \setminus \{ 0 \} \rightarrow \mathbb{C}$ be holomorphic. Show that the following are equivalent: There exists $g : \mathbb{C} \rightarrow \mathbb{C}$ holomorphic, such ...
strugglingStudent's user avatar
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1 answer
66 views

Analytic continuations for $(-1)^x$ over the real number line

The function $f(x)=(-1)^x$ can only be mapped to the real numbers when $x\in\mathbb{N}$. So I wonder what analytic continuation can expand the domain of $f$ to all real numbers without changing the ...
Kamal Saleh's user avatar
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Useful Partial Sums

The following formula: $$\sum_{k=m}^nf(k)=c(n-m)+\sum_{k=m}^\infty(f(k)-f(k+n))$$(Where $f\rightarrow c$) can be proven by telescoping the infinite sum in the RHS. The use of this formula is to expand ...
Kamal Saleh's user avatar
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can there be fractional values in the Veblen hierarchy?

I know that $\varphi_0(0)=\omega$ and $\varphi_1(0)=\epsilon_0$, but what would the value of $\varphi_{\frac{1}{2}}(0)$ be. it seems intuitive that it would be smaller than $\epsilon_0$, being $$\frac{...
Michael Toth's user avatar
2 votes
1 answer
85 views

How to find If Mellin transform has no poles?

I have the following integral: $$ F(s) = \int\limits_{\alpha}^{\delta} t^{s-3} f(t) dt. $$ The function $f(t)$ is real-valued, continuous, monotonically increasing on $[0,\delta]$, bounded, and is non-...
JavamonkYT's user avatar

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