Questions tagged [analytic-continuation]
For questions related to analytic continuation
300
questions
0
votes
0answers
11 views
Can $f(z)=\int_E \frac{dt}{t-z}$ be extended to an entire function?
Suppose that $E \subset \mathbb{R}$ is compact and $m(E)>0$. Let $\Omega=\mathbb{C} \setminus E$ and
\begin{equation}
f(z)=\int_E \frac{dt}{t-z}
\end{equation} for all $z \in \Omega$. I think that ...
0
votes
0answers
19 views
Extension of $f\in\mathcal{H}(\mathbb{D})$ that maps a circular arc to an analytic curve
Let $f\in\mathcal{H}(\mathbb{D})$ (i.e. it is a holomorphic function on the open unit disc) and $\gamma$ a (open) circular arc $\in \partial\mathbb{D}$ such that $f\in\mathcal{C}^0(\mathbb{D}\cup\...
0
votes
0answers
21 views
Analytic continuation of a holomorphic function $f$ is the same along any closed curve in $D$ implies $f$ extends to $D$?
Let $D\subset\mathbb{C}$ be a domain and let $f$ be the germ of a holomorphic function which admits unrestricted analytic continuation in some $E$ that contains $D$. Furthermore, suppose that analytic ...
0
votes
0answers
55 views
Why is analytic continuation practical?
I was exploring the Riemann Zetta function, and I observed that $\zeta (s)$ is not normally defined for $s $ such that $\Re (s) \leq 1$, but analytically continued to the whole complex plane. And ...
0
votes
0answers
27 views
analytic continuation of $\Phi(s)=\sum e^{-n^s}?$ [duplicate]
How do you analytically continue $\Phi(s)=\sum e^{-n^s}?$
I'm auditing complex analysis next fall but I'm curious about this function. I'm interested in this function because it converges and I ...
0
votes
0answers
22 views
Extending Local Function Germ to Global Morphism
I am wondering if the following argument is true.
Let $f\colon (\mathbb{F}^n,0)\to (\mathbb{F}^p,0)$ be a function germ, here $\mathbb{F}$ is either $\mathbb{R}$ or $\mathbb{C}$. For any pair of $\...
2
votes
1answer
59 views
Analytic continuation of $H_x^{(k)}=\sum_{n=1}^x \frac{1}{n^k}$?
Is there an analytic continuation of the generalised harmonic number $H_x^{(k)}=\sum_{n=1}^x \frac{1}{n^k}$ to the positive reals x, for $k>1$?
I can’t find anything useful through Google, just ...
0
votes
0answers
34 views
Is $\Phi$ the analytic continuation of $\Psi?$
Is $$\Phi(s)=\sum_n e^{-(n^s)}$$
the analytic continuation of
$$\Psi(s)=\sum_n (e^{-n^{-s}}-1)=\sum_{k\ge 1} \frac{(-1)^k}{k!} \zeta(sk)$$
which is analytic on $\Bbb C^∗−∪1/k$ because $(s−1)ζ(s)=O(...
0
votes
0answers
5 views
PDE and continuation method / Newton's method
I want to solve the nonlinear elliptic equation
$$
\begin{cases}
\Delta u+ f(u)=g \;\; \text{in} \; \; (0,1)\\
u(0)=u(1)=0
\end{cases}
$$
where $f\in C^1(\mathbb{R})$ and $g\in L^2(0,1)$
My idea,
...
1
vote
0answers
125 views
Analytic continuation of $\Phi(s)=\sum_{n \ge 1} e^{-n^s}$
While discussing theta functions, I thought:
$\zeta(s)=\sum n^{-s}=1+2^{-s}+3^{-s}+ \cdot\cdot\cdot$
$\Phi(s)=\sum e^{-(n^s)}=e^{-1}+e^{-(2^s)}+e^{(-3^s)}+\cdot\cdot\cdot $
What is the analytic ...
0
votes
1answer
42 views
How to show that the limit function fullfills $|F(z)| = 1$ for $|z|=1$ using the maximum principle?
Let $D\subset \mathbb{C}$ be bounded by $|z| = 1$ and some continuum $C$ in $|z| < 1$ such that $0 \not \in D$. The sequence $(f_n)$ of holomorphic and injective functions is defined on $D$ and ...
0
votes
0answers
21 views
The Definition of Analytic Continuation
I’ve been reading through Section 9.2 in Conways book on complex functions and I am a bit stuck on understanding Conway’s remark in the second to last paragraph on p.214. To be precise,
“That is, $$...
4
votes
0answers
108 views
Maximal extension of domain of $f(x)=\sum_{n=1}^\infty n^{\frac{1}{\log_n(x)}}$ using analytic continuation
Given $$ f(x)=\sum_{n=1}^\infty n^{\frac{1}{\log_n(x)}}=\sum_{n=1}^\infty e^{\frac{\ln^2(n)}{\ln(x)}} .$$
By inspection this is a sum of nonlinear hyperbolas, over $n$. It's because $\ln(x)\ln(y)=\ln^...
1
vote
3answers
114 views
Analytic continuation of a conformal map across the unit circle
I know that if $f$ is a conformal mapping of $\mathbb{D}$ onto some domain $D$ such that $\partial D$ is a Jordan curve, then $f$ has a continuous extension up to $\partial \mathbb{D}$ such that $f(\...
0
votes
1answer
37 views
Clarifications on a proof on analytic continuations
In the book Function of one complex variable of John B. Conway, the proof of this lemma on analytic continuations :
Lemma on Analytic Continuations
rest of the proof :
Proof of the preceding lemma
...
0
votes
0answers
13 views
Deducing domain of analyticity of the Laplace transform from a big O estimate: possible or not?
In e.g. the field of analytic number theory it seems quite normal to use transforms (for example the Laplace) to deduce results about asymptotics for certain counting functions, like in the prime ...
0
votes
0answers
32 views
On the continuation of the gamma function, $\Gamma(x)$ [duplicate]
So, on Wikipedia, it states that the gamma function, $\Gamma(z)$, $z\in\mathbb{C}$ and $\Re(z)\notin S$, for $S=\lbrace0,-1,-2,\cdots\rbrace$, is the analytic continuation of the integral
$$
I(z)=\...
2
votes
1answer
75 views
How to numerically perform analytic continuation?
I understand that one can in theory analytically continue a function by repeatedly computing new Taylor series. Suppose for example we have an analytic function $f$ defined on some open set $U$ and ...
0
votes
0answers
17 views
Analysis of the Adjoint representation of Lie Algebra
I'm attempting to find $f\left(ad_{x}\right)y$ where $ad_{x}y=xy-yx$ for any function f and rings x,y.
adjoint representation
I have a way to calculate it for any holomorphic function. However, I'm ...
0
votes
1answer
29 views
Analytic Continuation of Fractional Derivatives
I've been doing some work with fractional calculus, and I have been running into a problem. For a given meromorphic function $f: \mathbb{C}\to \mathbb{C}$, if a formula of the form $$(D^{\alpha}f)(0)=...
0
votes
0answers
68 views
On analytic continuation
If we have a power series, can we always get an analytic continuation of it? And if not, when can we? And is there a simple general method to get an analytic continuation of a divergent power series?
0
votes
0answers
91 views
How can one get an analytic continuation of a power series?
What method should one use to get an analytic continuation of this power series:
$$\sum_{n=0}^\infty \frac{x^n}{n\#}$$
; where $n\#$ is the product of all primes less or equal to $n$.
The radius of ...
7
votes
1answer
148 views
$f(z) = z + f(z^2)$ outside the unit disk?
The function $$
f(z) = \sum_{n=0}^\infty z^{2^n}
$$
which satisfies the functional equation $f(z) = z + f(z^2)$ is a classic example of a function analytic in $\mathbb{D} = \{z:|z|<1\}$ that cannot ...
0
votes
0answers
18 views
Analytic continuation of a multivariable function composition
Let $f\colon \mathbb{R}^n\to\mathbb{R}$ and, for all $i=1,\dots,n$, let $g_i\colon\mathbb{R}\to\mathbb{R}$. Then $f(g_1(x),\dots,g_n(x))$ maps $\mathbb{R}\to\mathbb{R}$.
Suppose that each $g_i$ has ...
0
votes
0answers
39 views
Do all values of $\zeta (s)$ follow from the Dirichlet series definition of $\zeta (s)$?
If we define $\zeta (s)$ by
$$\zeta (s)\overset{\text{def}}{=}\sum_{n\gt 0}n^{-s},\, \operatorname{Re}s\gt 1,$$
does it follow from the definition that e.g. $\zeta (-1)=-\frac{1}{12}$?
Or is it ...
0
votes
1answer
39 views
Meromorphic continuation of the Dedekind zeta function $\zeta_K(s)$ for all $s$ with $\sigma > 1 - \frac{1}{d}$.
Let $K$ be a number field of degree $n$ and $\mathcal{O}_K$ denote the ring of integers of $K$. For any complex number $s=\sigma +it$ with $\sigma > 1$, we define the Dedekind zeta function by the ...
1
vote
1answer
40 views
Meromorphic continuation of $\zeta(s)$ for all $s \in \mathbb{C}$ with $Re(s) > 0$
I am trying to obtain the meromorphic continuation of the Riemann zeta function $\zeta(s)$ for all $s \in \mathbb{C}$ with $Re(s) > 0$. Using the Abel's summation formula I've obtained that
$$
\...
-1
votes
1answer
60 views
The Riemann zeta function and analytic continuation, how many solutions are possible to his analytic continuation?
When Riemann developed the zeta function in the complex plane, he used a process called analytic continuation. He found only one solution. Does this mean that there was only one analytic solution to ...
0
votes
1answer
47 views
On a complex function $f(z)$ which is $\sqrt{z^2-1}\in\Bbb R $ when $x＞1$
Let $f(z)$ be a complex function whose value is $\sqrt{z^2-1}\in\Bbb R$ when $z\in \Bbb R$ is $＞１$, and let's suppose $f(z)$ is holomorphic if $1＜｜z｜＜\infty$.
I want to integrate $f(z)$ over $C$, a ...
1
vote
1answer
54 views
How to analytically continue the ‘two-dimension’ zeta function?
How to meromorphically continue
$$f(s)=\sum^\infty_{n=1}\sum^\infty_{m=1}\frac1{(an+bm)^s}$$ to $\Re s\le 2$? ($a,b$ are positive constants.)
In the case $a=b$, we have
$$f(s)=a^{-s} \sum^\infty_{n=1}...
0
votes
1answer
28 views
Asking for a reference which proves analytic continuation of Hypergeometric Function
I am self studying a research paper in analytic number theory and it requires analyticity of Hypergeometric Function .
I searched google but i couldnot found anything precize on how to prove ...
0
votes
0answers
25 views
What is the meaning of analytic continuation of $\Psi(z)$ when $z$ crosses the positive real axis from above?
I am reading an article A class of analytic perturbations for one-body Schrödinger Hamiltonians.
In the text there are expression like this
$...\Psi(z)$ can be analytically
continued when z crosses ...
1
vote
0answers
33 views
Doubt regarding proving analyticity of a power series
I am self studying concepts of complex analysis from Ponnusamy and Silvermann "Complex Variables with Applications"
In the chapter of analytic continuation in basic concepts authors mention that
...
0
votes
0answers
43 views
Properties of $\sum_{n=1}^{\infty} n^{-s\log n}$
I want to know about the function
$\displaystyle f(s):=\sum_{n=1}^{\infty} n^{-s\log n}$.
In particular, can this function be continued analytically around $s=0$?
0
votes
1answer
41 views
Do Divergent Integrals have a unique regularisation?
I know that the same question for divergent sums is false, but cannot find much on divergent integrals.
For example, consider the following divergent integral for positive reals $a,b$:
$\...
5
votes
1answer
141 views
A method to iterate the exponential function a non-integer number of times?
Notation
We employ the following notation:
$$ a_1 = e^{x} $$
$$ a_2 = e^{e^{x}} $$
$$ a_3 = e^{e^{e^{x}}} $$
$$ a_n = e^{\vdots^{e^{x}}}$$
We also use define $c$ by:
$$ e^c =c$$
Motivation
...
0
votes
1answer
71 views
Regarding applying concept of analytic continuation in Analytic number theory
I am studying analytic number theory from Tom M Apostol introduction to analytic number theory,
Actually Concept of analytic continuation was not taught by my instructor who took complex analysis ...
1
vote
0answers
86 views
Verifying uniqueness of my tetration
Previous two posts:
Numerical instability of an extended tetration
Verifying tetration properties
Update: The first link only verifies continuity on $\mathbb R$, and so continuity cannot be used for ...
1
vote
1answer
88 views
Let $f$ be a function which is continuous on the closed unit disc and analytic on the open disc.
Let $f$ be a function which is continuous on the closed unit disc and analytic on the open disc. Assume that $|f(z)| = 1$ whenever $|z| = 1$. Show that the function $f$ can be extended meromorphically ...
4
votes
0answers
45 views
Reference request for a book that covers analytic continuation in great detail starting from basics
I have earlier self studied Tom M Apostol Introduction to analytic number theory after doing a course in complex analysis, but my instructor at university didn't even mentioned analytic continuation, ...
4
votes
1answer
123 views
Continuation of functions beyond natural boundaries
The article Continuation of functions beyond natural boundaries by John L. Gammel states
I am particularly interested in the convergence of the $[N/N+1]$ Padé approximants beyond the natural ...
1
vote
1answer
60 views
Are all analytic continuations consistent with $1-1+1-1+\dots = \frac{1}{2}$?
Background:
This is a follow-up to this question, which is the same question except for a different series (and I suspect the answer may be different for this series—more on that later). In short, I ...
6
votes
0answers
75 views
Are all analytic continuations consistent with $1+2+3+\dots=-\frac{1}{12}$? [duplicate]
Background:
One way to "prove" that $1+2+3+\dots=-\frac{1}{12}$ is to define the function $f(z)=\sum_{n=0}^\infty \frac{1}{n^z}$ for $z\in\mathbb C, \operatorname{Re}(z)>1,$ then define $\zeta(z)$ ...
1
vote
0answers
59 views
Is it possible to deduce that $\mathbb C^2 \setminus \{ (0,0) \}$ is not affine directly from Hartogs's principle?
It is well-known that $X=\mathbb C^2 \setminus \{ (0,0) \}$ is not an affine variety. Perhaps the simplest proof of this fact consists of showing that every regular function on $X$ extends to a ...
1
vote
0answers
25 views
Integral involving a Gaussian and a fraction.
This question is a generalization of An identity involving the incomplete Beta function. .
Let $x\ge 0$ and $\epsilon_\pm \in (1,\infty)$. We consider the following integral:
\begin{equation}
{\...
8
votes
4answers
329 views
Is the “sum of all natural numbers” unique?
A while ago, there was a great hype about the “identity”
$$\sum_{n=1}^{\infty} n = -\frac{1}{12}.$$
Apart from some series manipulations where the validity seems to be at least questionable, the ...
1
vote
1answer
60 views
Analytic continuation in a proof in Apostol's analytic number theory textbook
In Apostol's Introduction to Analytic Number Theory, the following equation is derived for real $s>1$:
$$\Gamma(s)\zeta(s)=\int_0^{\infty}\frac{x^{s-1}}{e^x-1}\,\mathrm{d}x$$
Then, the fact that ...
0
votes
0answers
35 views
Proof of nonexistence of holomorphic continuation.
Let $D \subseteq \mathbb{C}$ be a domain and $\emptyset \neq U \subsetneq D$ open. I want to show the existence of a holomorphic function $f: U \rightarrow \mathbb{C}$ without holomorphic continuation ...
0
votes
0answers
15 views
Is a solution of an elliptic PDE constant if all of its derivatives vanish at a point? (in analogy to unique continuation principle)
Consider a one-dimensional elliptic second order differential equation on an interval $[a, b]$.
I am specifically interested in Sturm-Liouville problems where the principal symbol is the Laplacian and ...
1
vote
1answer
59 views
Analytic continuation of geometric series using Taylor expansion
I am currently trying to get a better grasp of the concept of analytic continuation and so I am working through this example:
Let $$f:(-1,1)\to\mathbb{R}~,~x\mapsto\sum_{k=0}^\infty x^k = \frac{1}{...
