Questions tagged [analytic-functions]

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4
votes
1answer
483 views

Existence of a smooth compactly supported function

Let $U$ be a bounded domain in $\mathbb R^n$. Does there exist a smooth function $f$ with compact support in $U$ such that: $$ \| f\|_{W^{k,\infty}(U)} \leq (k!)^{2-\epsilon},$$ for some $\epsilon>...
6
votes
1answer
537 views

Does it follow that $F^{(1)}(z)=F^{(2)}(z)$ for all $z \in \mathbb H$?

Let $\mathbb H= \{z \in \mathbb C\,:\, \textrm{Re}\,(z)\geq 0\}$ and for $j=1,2$ suppose that $F^{(j)}:\mathbb H\to \mathbb H$ is defined via $$ F^{(j)}(z) = \sum_{k=1}^{\infty} \frac{a_k^{(j)}}{z+\...
3
votes
2answers
415 views

In search for a counterexample related to the Abel-Stolz theorem

Disclaimer: I posted this question seven days ago here on the Math.SE, with slightly different (however in an inessential way) comments. The question has been upvoted but no answer has been given, so ...
1
vote
0answers
98 views

A problem related to analytic function

Let, $z,w\in \mathbb{C}$. Let, $f(z)$ be an analytic function in $|z|<1$. Define, $f(z)= g(w)$ where $g(w)$ is analytic function in $\Re(w)>1/2$ and $w=\frac{1}{1+z^2}$ . Question Prove that $$\...
6
votes
1answer
225 views

Real analyticity of continuous function via restriction to analytic curves

Suppose $X\subset \mathbb R^n$ is an irreducible real analytic sub-variety (i.e. the set of solutions of a system $f_1=\ldots=f_k=0$ with $f_i$ analytic) Let $x\in X$ be a point and let $F: X\to \...
2
votes
0answers
76 views

An analytic function, asymptotically expandable in a Dirichlet series, is the sum of this series

Let there be a function $F(s)$ that is analytic in some half-plane $\sigma>\sigma_0$ (where $s=\sigma + it $). Let the function $F(s)$ have an asymptotic expansion of the form $F(s)\sim\sum\limits_{...
3
votes
0answers
88 views

Properness of real analytic maps?

Fix a polynomial mapping $\mathbb R^n\overset{f}{\to} \mathbb R$. This answer shows that if the top degree homogeneous component of $f$ is zero only at the origin, then $f$ is proper. Intuitively, ...
1
vote
2answers
217 views

Global theory of holomorphic functions [closed]

I am trying to develop a theory explaining analytic continuation of a holomorphic function $f(z)$ defined on an open set $D \subset \mathbb{C}$. Recently, I was looking at the last chapter of Lars ...
1
vote
0answers
139 views

Is a mixture of real analytic functions again analytic?

Let $$h : \mathbb{R}^2 \to \mathbb{R}^+.$$ Suppose that for each $x$, $h(x, y)$ is a real analytic function of $y$. Let $\mu(dx)$ be a finite measure on $\mathbb{R}$, and for each $y$, suppose that $$...
3
votes
1answer
185 views

Looking for a sequence of analytic functions with strange behaviour

Let $K_1 \subsetneq K_2$ be two non-empty compact sets and let $D = (d_n)_{n \in \mathbb{N}}$ be a dense sequence on $K_2\smallsetminus K_1.$ Consider $f_n : \mathbb{C}\smallsetminus K_1 \rightarrow \...
4
votes
1answer
291 views

Analytic functions in arbitrary rings?

We have developed a rich theory of analytic functions over $\mathbb{R}^n$ and $\mathbb{C}^n$. This is pretty reasonable, as analyticity here (local representation by power series) is closely linked to ...
0
votes
0answers
36 views

Analyticity of $ \int\limits_x {\int\limits_y {Q(a\Re (x + y))Q(b\Im (x + y))\log \left\{ {Q(a\Re (x + y))Q(b\Im (x + y))} \right\}d{F_x}d{F_y}}}$

I need to show that the following function is analytic on the bounded complex plane. Lest define the function, $f = \int\limits_x {\int\limits_y {Q(a\Re (x + y))Q(b\Im (x + y))\log \left\{ {Q(a\Re (x +...
3
votes
1answer
66 views

Analyticity of $f = Q(a\Re (x + y))Q(b\Im (x + y))\log \left\{ {Q(a\Re (x + y))Q(b\Im (x + y))} \right\}$ in the complex plane?

Let I have the following function, $f = Q(a\Re (x + y))Q(b\Im (x + y))\log \left\{ {Q(a\Re (x + y))Q(b\Im (x + y))} \right\}$ Where, $x,y \in C$, $a,b\in R$ and $- m \le \Re (x),\Re (y),\Im (x),\Im (y)...
1
vote
0answers
78 views

A Riemann Hilbert problem on the unit square

Let $p_0=0$, $p_1=1$, $p_2=1+i$ and $p_3=i$ be the four vertices of a square $Q$ on the complex plane $\mathbb C$. Let $f \in C^{\infty}_c((0,1))$ and consider the following Riemann-Hilbert problem on ...
6
votes
1answer
332 views

A smooth function $\mathbb{R}\to\mathbb{R}$ agrees with an analytic function on a bounded infinite set

Fix a smooth function $f:\mathbb{R}\to\mathbb{R}$. Do there exist real numbers $a<b$, an infinite set $S\subset (a, b)$ and an analytic function $g$ defined on $(a-\epsilon, b+\epsilon)$ for some $\...
14
votes
0answers
666 views

Lower bounds on analytic functions connected to Fox H

The question is related to the one I asked before and never got an answer to. Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$ . I need to demonstrate that the ...
5
votes
3answers
516 views

real analytic function with given shape

I am looking for a 5 parameter family of analytic functions $f:[0,1]\to \mathbb{R}$ such that (0) $f$ has zeros at $0,p,1$. (1) $f$ is convex in $[0,p]$ and concave in $[p,1]$. (2) The five parameters,...
1
vote
0answers
40 views

Independence of variables in curvilinear coordinate systems

Let $U$ be a connected open subset of $\Bbb{R}^n$, and let $(\xi_1,\dots,\xi_n)$ be a curvilinear smooth ($C^\infty$) coordinate system on $U$. Suppose $1\leq k<n$. A smooth function $f:U\...
4
votes
1answer
171 views

Smoothness of the radius of convergence

Let $(x\mapsto a_n(x))_n$ be a sequence of smooth functions defined on some fixed interval $I$. Consider the power series $\sum_{n\geq 0}a_n(x)t^n$ and denote by $R(x)$ its radius of convergence. Does ...
1
vote
0answers
31 views

Analytic function whose derivatives and primitives are independent from a given set of countable cardinality

Let $L=(l_j)_{j\in\mathbb{N}}$ be a set of countably many independent real analytic functions on $[0,2\pi]$. Here and in the following, independent means that a function cannot be written as finite ...
1
vote
1answer
205 views

Constraint from level sets of an analytic function

Let $\theta$ and $f$ be two real analytic non-constant functions defined on $[0,2\pi]$. For simplicity we assume $f$ has just two critical values $m<M$ (in the picture $-1$ and $1$); we index as $\{...
-3
votes
1answer
187 views

Conformal map from a 7-sided polyhedron to a square pyramid

I have a right-angled square pyramid, $A$, whose height and base-length are $l$. By 'right-angled', I mean that the apex of $A$ lies vertically above one of the vertices in its base. Now supposed I ...
4
votes
2answers
318 views

Angle of analyticity of semigroup

Is there any known parabolic PDEs in the literature where the angle of analyticity of the associated semigroup is $<\pi/2$ ? For example, the angle of heat semigroup in $L^2$ is exactly $=\pi/2$. ...
1
vote
0answers
65 views

Analytic continuation of function of two complex variables

Consider $f(z_1, z_2)$ a function of two complex variables, symmetric in its arguments $z_1$ and $z_2$. Consider the regions: \begin{eqnarray} \mathcal{R}_1= \{ (z_1, z_2) \in \mathbb{C}^2: Re(z_1)>...
3
votes
0answers
81 views

Is $|f^{-1}f(p)|$ constant on a conull set?

Suppose $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is an analytic map. Can we say that there is a conull set $U \subset \mathbb{R}^n$ (i.e. $\mathbb{R}^n \setminus U$ has measure zero) where $|f^{-1}f(...
0
votes
1answer
258 views

Fubini/Tonelli theorems for expectation of power series

as part of a proof in a paper i have statement, i cannot figure out how to proof: Assume $(c_k)_{k\in \mathbb{N}}$ is a sequence of nonnegative random variables and $g: (-1,1] \to \mathbb{R}$ is a ...
2
votes
1answer
150 views

Possible condition for a many variable holomorphic map to be locally surjective

Suppose $a \in \mathbb C^n$, $U$ is a neighbourhood of $a$, and $f: U \to \mathbb C^n$ is analytic. Let $b = f(a)$ and suppose also that $f^{-1}(b) = \{a\}$. Must the image of $f$ contain a ...
2
votes
1answer
721 views

Essential singularity [closed]

In shaum's outline complex analysis,definition of essential point is: An isolated singularity that is not pole or removable singularity is called essential singularity Now in the same book there is an ...
9
votes
0answers
448 views

On Riesz criteria for Riemann hypothesis:

Marcel Riesz defined a function : $R(x) = \sum_{n=1}^\infty \frac {(-1)^n x^n} {\zeta(2n)\Gamma(n)}$ The Riemann hypothesis holds if $R(x)= O( x^{1/4 + {\varepsilon}}$) For any $\varepsilon$ We have ...
5
votes
2answers
200 views

Intuitive explanation of regularized products

I've come across some regularized product during study of zeta regularization . We can prove various results like : $ \infty != \prod_{k=1}^\infty k = \sqrt{2\pi} $ I also know the proof using $\...
7
votes
0answers
427 views

On a paper of Alain Connes entitled 'Around Wilson's Theorem '

A relatively recent paper Alain Connes - Around Wilson's theorem introduced the function $$ S(n,x ) = \sum_{i=1}^n \sin^2\Bigl(\frac{(i-1)! x}{i}\Bigr). $$ In the same paper, he proved that the ...
6
votes
0answers
235 views

Complex factorization of the angular part of the Laplacian

Some time ago some research led me to the following equality: \begin{equation} \frac{1}{\sin^2 \phi }\frac{\partial^2 }{\partial \theta^2} +\frac{\partial^2 }{\partial \phi^2} +\cot \phi \frac{\...
1
vote
1answer
95 views

On a case of real-analytic interpolation

Given strictly increasing sequence $x_n$ of rational numbers with $\sup x_n = x$. In which case (sufficient condition on $x_n$) there exists real-analytic function $f:U_\epsilon(0)\to\mathbb{R}$ ...
8
votes
1answer
246 views

Non-normal numbers definable without parameters in the langauge of differential rings with composition

Background: It is currently unknown whether $e$ is normal. A natural way to approach this question is to find a class to which $e$ belongs, and prove all members of that class are normal. For example, ...
17
votes
2answers
785 views

When is $\sum_{n\in\mathbb Z} f(x+n)$ constant?

A recently asked question (linked here) deals with the remarkable identity $$ \sum_{n\in\mathbb Z} \mathrm{sinc}(n+x)= \pi,\quad x\in\mathbb R, $$ where $\mathrm{sinc}(x)=\sin(x)/x$. It is easy ...
2
votes
2answers
162 views

One-Sided Analyticity Condition Guarantees Analytic Function?

Let $f \ \colon \ [0,\infty) \to \mathbb{R}$ be a function satisfying: $f$ is differentiable infinitely many times in $(0,\infty)$, and has a right-derivative of any order at $0$. $f$ satifsfies the ...
1
vote
1answer
297 views

Steepest descent integration in several dimensions

The method of steepest descent provides an asymptotic approximation for integrals of the form: $$I = \int_C \exp(M f(z))\mathrm dz$$ for large positive $M$, where $f(z)$ is analytic in the region of ...
11
votes
2answers
626 views

Are conformal maps between Riemannian manifolds real-analytic?

This is a cross-post. Let $M,N$ be oriented smooth ($C^{\infty}$) $n$-dimensional Riemannian manifolds, and let $f:M \to N$ be a smooth orientation-preserving weakly* conformal map. Do there exist ...
4
votes
1answer
233 views

An inequality of T. Carleman

I'm looking for the name and some references for the proof of the inequality below. I founded that is due to T. Carleman but no reference was given. Let $f(z)$ be an analytic function on a subdomain $...
11
votes
1answer
881 views

Dual of the space of all bounded holomorphic functions

Let $\mathbb{B}$ be the open unit ball in $\mathbb{C}^n, n\geq 1$ and let $H^\infty (\mathbb{B})$ be the space of all bounded holomorphic functions on $\mathbb{B}$. It is well known that $H^\infty (\...
1
vote
0answers
148 views

non-analytic functions with arbitrary large derivatives [closed]

This may be a trivial question but I can't see it immediately. Suppose $\{a_k\}$ is an increasing sequence of positive reals. Does there exist a smooth function $f \in C^{\infty}([0,1])$ such that $\...
1
vote
0answers
101 views

Is the topology generated by the complements of analytic subsets strictly coarser than the Euclidean topology in dimensions $\geq 2$?

Let $\mathbb{K}$ be either $\mathbb{R}$ or $\mathbb{C}$ and let $N\geq 2$. Similarly to the construction of the Zariski topology, take the collection of zero sets of $\mathbb{K}$-analytic functions to ...
5
votes
2answers
1k views

Bounds on the number of zeros of real analytic functions

Let $F(A)$ be a class of real-analytic function on an interval $A \subset \mathbb{R}$ minus the zero function. We have the following theorem for $F(A)$. If $f \in F(A)$ then $f$ has at most ...
0
votes
0answers
186 views

Sequence of analytic functions

Let $f_k$ be a sequence of rational functions analytic in the discs $\{ |z| < 1 + \epsilon_k\}$ (with some $\epsilon_k > 0$), which converge to an analytic function $f$ in every point $|z| < ...
0
votes
1answer
104 views

Elegant / Canonical way to Extend Integer Iterates of a Function to a Real Parameter

Any map $f \colon \mathbb{R} \to \mathbb{R}$ induces a "composition map" $$f^\circ\colon \mathbb{R} \times \mathbb{N} \to \mathbb{R},$$ where $$f^{\circ n}(x) = \underbrace{f \circ \dotsb \circ f}_{...
6
votes
1answer
341 views

"Descent" of analytic functions along a finite morphism

Let $f : X \to Y$ be a finite surjective morphism of smooth affine algebraic varieties over the complex numbers. Is it true that a function on $Y$ whose pullback via $f$ is an analytic function on $X$,...
1
vote
2answers
198 views

Numerical evaluation of some series

Let $k\geq 1$ be an integer and let $P(n)$ be the polynomial $\binom{n+k}{k}$. Consider the series $$ L_k(s) = \sum_{n \geq 0} \frac{P'(n)}{P(n)^s}. $$ It is known (by previous work of myself and ...
6
votes
2answers
424 views

Critical values of analytic functions of several variables

Let $f:\mathbb{R}^d\to \mathbb{R}$ be real analytic. Define $S=\{x\in\mathbb{R}^d, \nabla f (x)=0\} $. Is it true that for any compact set $K\subset \mathbb{R}^d$, $f(S\cap K)$ is a finite set ?
2
votes
1answer
157 views

If $Z$ is standard normal and $f$ is analytic. Is $g(t)= E[ f(Z-t)]$ analytic?

Let $Z$ be a standard normal. Now define \begin{align} g(t)= E[ f(Z-t)] \end{align} where $f(x)$ is a real-analytic function and $|f(x)| \le x^4$. Question: Is it true that $g(t)$ is also a real ...
5
votes
1answer
268 views

Are continuous rational functions arc-analytic?

Let $X\subseteq\mathbb{R}^n$ be a smooth semi-algebraic set (for simplicity we can assume $X=B(0,r)$ is a small ball around the origin). A function $f:X\rightarrow \mathbb{R}$ is called a continuous ...