Questions tagged [analytic-continuation]
For questions related to analytic continuation
266
questions with no upvoted or accepted answers
8
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0
answers
79
views
A Family of Limits Leading to an Interesting Function
A while back I got very interested in limits of the form
$$
\lim_{n\to\infty} (2A)^n \left (A-\underbrace{\sqrt{a+\sqrt{a+\ldots\sqrt{a+z}}}}_{n\textrm{ radicals}} \right )=f_a^{-1}(z)
$$
Where $A$ ...
8
votes
0
answers
659
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 ...
7
votes
0
answers
556
views
Integral representation for series of any order
Hello infinity series enjoyers. If my calulations are correct, I think I made some integral representation of any divergent and convergent seris of any order s. All calulations and the question are ...
6
votes
1
answer
428
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 ...
5
votes
0
answers
702
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 ...
5
votes
0
answers
347
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 ...
5
votes
0
answers
119
views
Analyticity of determinant formula for Gaussian integral
It is a well known fact that $\int_{\mathbb{R}^n} e^{-\frac{1}{2}x \cdot A x} dx = \sqrt{\frac{(2\pi)^n}{\det{A}}}$ for real, positive definite $A$. The left hand side of the equation make sense for ...
5
votes
0
answers
270
views
Nevanlinna - Herglotz - Pick - R functions : boundary values on the real axis
Nevanlinna/Herglotz functions are analytic functions defined on the upper half complex plane and have non-negative imaginary part there.
A remarkable theorem asserts that every such function has a ...
5
votes
0
answers
366
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 ...
5
votes
0
answers
344
views
What is the relationship between the asymptotic series outside the radius of convergence and the analytic continuation?
In his post about computing the zeta function by smooth cutoff regularization of the series $\sum\frac{1}{n^2}$, Terry Tao shows how $-{1\over 12}$ is the finite part of the asymptotic series for the ...
5
votes
0
answers
250
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
votes
0
answers
261
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 (...
4
votes
0
answers
584
views
Lipschitz functions (Theorem 1.4 of Condenser Capacities and Symmetrization in Geometric Function Theory)
I am struggling to understand the second part of the following proof of Theorem 1.4 of the book "Condenser Capacities and Symmetrization in Geometric Function Theory" by Vladimir N. Dubinin (...
4
votes
0
answers
110
views
Does a smooth cutoff function analytically continue any function along the real line?
I have been trying to prove or disprove for a few weeks (to avail) whether or not multiplying by a sufficiently smooth cutoff will analytically continue a function up until it hits a singularity on ...
4
votes
0
answers
90
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
votes
0
answers
293
views
Riemann zeta meromorphic cont. using Abel summation formula
In Stein&Shakarchi, Complex Analysis, chapter 6, problem 2-3 (p. 180), they hint at a method to meromorphically continue the zeta function to the entire complex plane. I can see from Abel's ...
4
votes
0
answers
113
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
votes
0
answers
492
views
Comparison of entire functions with $f=cg$
Let $f,g:\mathbb C\to\mathbb C$ be holomorphic with $|f(z)|\leq|g(z)|$ for all $z\in\mathbb C$. Show that there is a $c\in\mathbb C$ with $|c|\leq 1$ such that $f=cg$.
If $g\equiv 0$ then we ...
4
votes
0
answers
56
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 ...
4
votes
0
answers
320
views
Analytic continuation of an integral involving the Mittag-Leffler function
I have posted this question on MO, and I didn't get an answer. We have the following integral:
$$I(s)=\int_{0}^{\infty} \frac{s}{2x}\left(E_{s/2}((\pi x)^{s/2})-1\right)\omega(x)dx -\lim_{z \to 1 }\...
3
votes
0
answers
51
views
Computing ratio of two sums
I'm interested in computing
$$ f(s) := \frac{ \sum_{n=0}^{\infty} a_n s^n } { \sum_{n=0}^{\infty} b_n s^n } $$
for some given $s \in \mathbb{C}$, where the power series in the numerator and ...
3
votes
0
answers
163
views
conformal mapping from unit disk to equilateral triangle
Let $$f(z) = C \int_{0}^{z} \frac{1}{(1-\zeta^3)^{\frac{2}{3}}}d\zeta$$ with $$C \int_{0}^{1} \frac{1}{(1-x^3)^{\frac{2}{3}}}dx=1$$ be a conformal map from the unit disc $B(0,1)$ to the equilateral ...
3
votes
0
answers
159
views
Explicit Taylor Expansions of Zeta Function
If one's goal were to calculate one value of the zeta function, say $\zeta(-1)$ is at all feasible to do this explicitly by calculating Taylor series and doing an analytic continuation "by hand&...
3
votes
0
answers
201
views
$\sum\limits_{n=1}^\infty\frac{(-1)^\frac{(5+\sqrt3i)n}6x^n}{n!}(\frac ns )^{(n-1)}\,_2\text F_1\left(1-n,-sn;2-sn;s\right)$ closed form or integral.
From:
Completing the explicit $\lim\limits_{c\to0}\text I^{-1}_{cx}(r,c)$/ inverse $\int_0^x\frac{t^{r-1}}{1-t}dt,r\in\Bbb Q$ series expansion
here is a Lagrange inversion expansion with $r=\frac23$ ...
3
votes
0
answers
82
views
Inverse and extension of the $n$th Airy Ai and Bi zero
$\def\ai{\operatorname{ai}} \def\bi{\operatorname{bi}} \def\Ai{\operatorname{Ai}} \def\Bi{\operatorname{Bi}} $ Airy Ai Zero $\ai_n$ gives the $n$th zero of the Airy Ai function and Airy Bi Zero $\bi_n$...
3
votes
0
answers
202
views
Does this function come up anywhere in mathematics? $f(s)=\sum_{n=0}^\infty\frac{(-1)^n}{n!}\zeta(-ns)$
I'm wondering if anyone knows if this function comes up anywhere in mathematics:
$$f(s)=\sum_{n=0}^\infty\frac{(-1)^n}{n!}\zeta(-ns)$$
where $\zeta(s)$ is the Riemann Zeta function.
I'm asking because ...
3
votes
0
answers
106
views
Analytic continuation of $g(\sqrt{1-x^2})$ where $g$ is entire
Consider a function $f(x)=\sqrt{1-x^2}, x\in \mathbb{R}$. Moreover, let $g:\mathbb{C} \to \mathbb{C}$ be an entire function and consider the composition
\begin{align}
h(x)= g(f(x))=g(\sqrt{1-x^2})
\...
3
votes
0
answers
61
views
Proving two polynomials converge to the same function
I am looking to create polynomials that converge past the usual radius of convergence on the real line. So far, I have proved that this polynomial will converge to the analytic continuation of the ...
3
votes
0
answers
82
views
Can the Bernoulli numbers be viewed as a 'renormalization' of a finite geometric series with term $e^{-x}$, by integrating over $(-1,0)$?
I was playing around with ways to calculate Bernoulli numbers; for this post I will take their generating function as $x/(1-e^{-x})$, that is,
$$
\sum_{n=0}^{\infty} \frac{B_n}{n!}x^n = \frac{x}{1-e^{-...
3
votes
0
answers
525
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 ...
3
votes
0
answers
334
views
maximal analytic continuation of a holomorphic germ on riemann surface
This is related to Forster, Riemann surfaces Exercise 7.1
Let $X$ be a riemann surface and $a\in X$ be a point with $\phi\in O_{X,a}$ where $O_{X,a}$ is the set of holomorphic germs at $a$. Let $Y$ ...
3
votes
0
answers
252
views
Abel-Plana Analytic Continuation of the Partial Sums of the Zeta Function
So I am trying to analytically continue the formula for the partial sums of the Riemann zeta function by using the fact that, if $\zeta_{k}$ represents the $k^{\text{th}}$ partial sum of the zeta ...
2
votes
0
answers
39
views
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 ...
2
votes
0
answers
54
views
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 ...
2
votes
0
answers
85
views
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 $Y^{d+1}:$
$$ \zeta_1( t)=\bigg(\lim_{k \to\infty}\bigg( \int_{Y}\sum_{n=1}^k\varphi_n(x)dx\bigg)^{-1}\int_{X} \sum_{n=1}^k \...
2
votes
0
answers
16
views
Dense subspace for arbitrary powers of a positive operator
Let $\mathcal{H}$ be a Hilbert space, and let $\Delta$ be a positive, self-adjoint, unbounded operator. I think I can prove the following statement, but I find it quite surprising, and I can't find it ...
2
votes
0
answers
37
views
Analytic continuation of integral representation as Cauchy principal value
Suppose I have the following formula,
\begin{equation}
\mathrm{P}\int_{-\infty}^{\infty} f(x,q) dx = F(q),
\end{equation}
for all $q\in\mathbb{R}$, where P stands for the Cauchy principal value.
If $F(...
2
votes
0
answers
33
views
The relationship between interior and exterior Riemann maps
Let $U\subset\widehat{\mathbb{C}}$ be a compact Jordan domain with $\partial U$ connected. Is there a relationship between the exterior Riemann map associated to $U$,
$$\varphi:\mathring{(\mathbb{D}^c)...
2
votes
0
answers
51
views
Extending functions from integers to reals. What properties (generally) matter?
There's no shortage of functions $f:\mathbb{Z}\to\mathbb{R}$, or perhaps more generally $f:\mathbb{N}\to\mathbb{C}$, which then later get defined over some continuum like $g:\mathbb{R}\to\mathbb{R}$ ...
2
votes
0
answers
84
views
Is analytic continuation the same as solving a PDE boundary problem with Cauchy Riemann
I was considering the idea of analytic continuation:
Let $ U \subseteq \mathbb{R}$ be a subinterval (perhaps all of R). Then suppose we have some smooth function defined $f: U \rightarrow \mathbb{R}$ ...
2
votes
0
answers
123
views
When will a Taylor series analytically continue a holomorphic function?
Suppose we have a complex-analytic function $f$ given by
\begin{equation}
\tag{1}
f(z) = \sum_{n=0}^{\infty}\, a_n \mspace{.5mu} (z-z_0)^n, \qquad a_n \in \mathbb{C},
\end{equation}
where $z_0$ is a ...
2
votes
0
answers
55
views
How are analytic continuations computed?
Suppose I define some analytic function on some domain $\Omega_1$ and I want to extend this function to the unique analytic continuation on $\Omega_2$. Are there any general methods of computing the ...
2
votes
0
answers
74
views
How can I calculate an analytic continuation of this power series
This is probably an easy/trivial question, but being an engineer my background on maths isn't very strong and I'm unable to find the solution to this problem. I hope anyone can help.
I have been able ...
2
votes
0
answers
81
views
Question on analytic continuation of functions related to $\log\zeta(s)$ and $\frac{\zeta'}{\zeta}(s)$ that converge for $|\Re(s)|>1$
Consider the following two Dirichlet series
$$C_3(s)=\underset{N\to\infty}{\text{lim}}\left(\sum\limits_{n=1}^N 1_{n=p^k}\, \log(n)\, n^{-s}\right),\quad\Re(s)>1\tag{1}$$
$$K_\Omega(s)=\underset{N\...
2
votes
0
answers
72
views
Description of complete analytic function for $\sqrt{4z-\sqrt[3]{z}}$.
The problem is to describe all branches and all the curves of analytic continuation of $\sqrt{4z-\sqrt[3]{z}}$.
I started with representing function in a way $\sqrt{4w^3-w}\circ\sqrt[3]{z}$
So, there ...
2
votes
0
answers
107
views
trying to understand analytic continuation (2 scenarios)
The following two scenarios I hope will help me fill in gaps in my understanding of analytic continuation.
Scenario 1
We have a function $f(z)$ of a complex variable $z$. It is known to be zero at $z=...
2
votes
0
answers
73
views
Finding the best Taylor-esque polynomial
Background
Based only on the information provided by derivatives at a single point, the Taylor series provides the 'best'* approximation of the function around its radius of convergence. But, some ...
2
votes
0
answers
76
views
Analytic Continuation of Fractions
Excuse my lack of expertise, I study natural sciences (physics) and not mathematics so I will be off with my terminology and mathematical vocabulary. Please feel free to poke and build at this idea ...
2
votes
0
answers
74
views
Question about $\sum_{k=0}^\infty \frac{1}{(H_{2^k})^x}$
Define a function:
$$f(x)=\sum_{k=0}^\infty \frac{1}{\bigg(\sum_{n=1}^{{2^k}}\frac{1}{n}\bigg)^x}=\sum_{k=0}^\infty \frac{1}{(H_{2^k})^x} $$
where $H_k$ is the k-th harmonic number.
Questions:
For ...
2
votes
0
answers
81
views
Analytic continuation of a function on the right half-plane to a region enclosing the circle $\{ \lvert z \rvert = 3\}$
I am trying to solve the following question:
Show that there exists an analytic function $f$ in the open right half-plane such that $(f(z))^2 + 2f(z) \equiv z^2$. Show that your function $f$ can be ...