The analytic-functions tag has no usage guidance.

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### Zeroes of trigonometric-like function

Consider a function $f(z)=\cos(z)\cosh(az)+\sin(z)\sinh(bz)$ for $z\in \mathbb{Z}, a,b \in \mathbb{R}$. Denote $D\subseteq \mathbb{R}^2$ being the set of such pairs $(a,b)$ of parameters so that NOT ...

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### A proper class for smooth chaotic function

This might be a little, soft, but I'll try
Consider the interval $I=[-1,1]$. We will define a chaotic function $f:\mathbb{R}_+ \times I \to \mathbb{C}$ in the following traditional way:
For every ...

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40 views

### Modulus of Continuity for an Analytic Function on an Ellipse

This is a question which I stumbled upon while working on Legendre Polynomials, but it is actually a question in complex analysis.
Consider: Given $f\in C^{\infty} (E)$, where $E_{\rho} \subseteq ...

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55 views

### Separating the points of projective spaces with real-analytic functions

Is there an easy way to separate the points of $\Bbb C \Bbb P^n$ or $\Bbb R \Bbb P^n$ (viewed as real-analytic manifolds) with real-analytic functions? If two points lie in a coordinate patch where a ...

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107 views

### Does the Abel transform preserve analyticity?

Let $I=(0,1]$ and $T=\{(x,y)\in I^2;x\geq y\}$.
If functions $f:I\to\mathbb R$ and $w:T\to\mathbb R$ are analytic, is the function $A_wf:I\to\mathbb R$,
$$
...

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308 views

### Are the zeroes of the Fourier Transform of compactly supported functions isolated?

I have a continuous function $f$ on a locally compact Abelian group $G$ with compact support, and I would like to say that the zeroes of $f$ are sparse in some sense (isolated would be good, uniformly ...

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149 views

### Improving Baumgartner's result?

Q1: Is it consistent with the failure of CH to have an $\aleph_1$-dense subset $A \subseteq \mathbb{R}$ such that for every $X \subseteq \mathbb{R}$ of size $\aleph_1$, there is a $C^{\infty}$ map $F: ...

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71 views

### Composition algebra of Gevrey function for $s<1$

Let $g,f$ be real-valued functions defined on the real line. Let $s$ be a real number.
Assuming that $g,f$ are both in the Gevrey class $G^{s}$, it is true that $g\circ f$ belongs to $G^{s}$ if $s\ge ...

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### Do any specializations of variables give valid equalities of series and products involving Witt symmetric functions?

Formally, Witt symmetric functions $w_n(x_1,x_2,...)$ ($n\geqslant1$) can be defined by
$$
\prod_n(1-w_nt^n)=1+\sum_k(-1)^ke_kt^k=\prod_j(1-x_jt),
$$
where $e_k(x_1,x_2,...)$ are the elementary ...

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226 views

### Extensions of Real Analytic to Holomorphic Functions in One & Several Variables: References?

A problem I'm working on requires the application of Cauchy's estimate for the modulus of the coefficients of a holomorphic function's power series representation, but the original functions with ...

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707 views

### A question of Erdos on entire functions

At the end of the following paper, Erdos asked if there is a family $F$ of entire functions of size continuum such that for every $z \in \mathbb{C}$, $\{f(z) : f \in F\}$ has size less than continuum. ...

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258 views

### Does there exist a nonconstant, periodic, real analytic function with period 1 and rational Maclaurin coefficients?

Does there exist a nonconstant, real analytic function $f \colon \mathbb{R} \to \mathbb{R}$ such that $f$ is periodic with period 1 and whose Maclaurin coefficients are all rational?
(The function ...

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132 views

### Interesting property of analytic functions

Let $f:(t_0-\varepsilon, t_0+\varepsilon)\to\mathbb{C}$, be an analytic application, such that: $f(t)=0\Longleftrightarrow\ t=t_0$.
Is it true that there is an analytic function $g:(t_0-\varepsilon, ...

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294 views

### What function space does holomorphic functional calculus give us?

Let $A$ be a unital Banach algebra, $U$ be an open subset of $\mathbb{C}$, and $A_U:=\{x\in A:\sigma(x)\subset U\}$. Holomorphic functional calculus says that any holomorphic function ...

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### Existence of zero-free strip of a Laplace transform (edited ..)

Problem
Let $\beta$ be a probability measure on $\mathbb{R}$, and define
$$
K = \left \{z \in \mathbb{C}: g\left(z\right)=\int_{-\infty}^{\infty}\exp\left(z x\right)\beta ( dx ) \text{ is ...

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112 views

### Complements of images of complex analytic sets

It is known that the complement of an analytic set is connected. In general, the complement of a proper complex analytic set in a connected complex manifold is an arcwise connected dense open set. My ...

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206 views

### analytic vector bundles

Let $E$ be a real analytic vector bundle on an analytic manifold $M$. Assume that $E$, as a smooth vector bundle, is a trivial bundle.
Is $E$ a trivial analytic vector bundle?
I need to the ...

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124 views

### Analytic version of the Cartan lemma

Assume that $\beta$ is a real analytic 2-form on an analytic manifold $M$ and $\alpha$ is an analytic non vanishing 1-form on $M$. Assume that $\beta \wedge \alpha=0$. Is there an analytic 1-form ...

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241 views

### A cohomology associated to a 1- form

In this question all objects are real analytic.(manifolds, differential forms..)
Assume that $M$ is a compact manifold and $\alpha \in \Omega^{1}(M)$ is a one form.
We define a map ...

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511 views

### I don't understand behavior of this integral, help!

In an answer to a question I needed the following integral:
$$
f(z):=\int\limits_0^\infty t\coth(zt)e^{-t^2}dt;
$$
it represented deviation from modularity of some other function. However I noticed ...

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130 views

### Comparison of two infinite dimensional Lie Algebras

Is there an example of a real analytic (compact) manifold $M$ such that the following two lie algebras are isomorphic Lie algebras:
$\chi^{\infty}(M)$, the Lie algebra of all smooth vector ...

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62 views

### hyperfunctions and analytic duals

Let $A(\mathbb R^n)$ be the real analytic functions and $\mathscr B(\mathbb R^n)$ the hyperfunctions, dual to $A(\mathbb R^n)$. Further let $W\subset \mathbb C$ be a cone in the complex plane with ...

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### Derivation on real analytic manifolds

Let $M$ be a real analytic manifold. By $C^{\omega}(M)$ we mean the algebra of all analytic functions from $M$ to $\mathbb{R}$. Assume that $D$ is a derivation on $C^{\omega}(M)$ .
Is there a ...

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195 views

### Real analytic ergodic diffeomorphisms of the two sphere

Does there exists a real analytic area preserving ergodic diffeomorphism on $S^2$?
(Possibly by perturbing a rotation in the real-analytic topology?)

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### Example of n-parameter family of real-analytic diffeomorphisms acting on $S^3$, constant on the Hopf fibres

I am trying to construct an n-parameter family of measure preserving real analytic diffeomorphisms on $S^3$ which preserves the $S^1$ fibres of the Hopf fibration but acts transitively on the image ...

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87 views

### Lagrangean equations for the generating function of quadrangulations

Let $M(z)$ be the generating function of edge-rooted connected quadrangulations, with $z$ marking the number of edges. I derived the following Lagrangean equations for $M(z)$:
$$M(z) = ...

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### Examples of Bivariate Analytic, Globally Continuous Functions, whose Set of Zeros are Boundaries of Polygons

Are there any known examples of analytic, globally continuous functions $f(x,y): (x,y)\in\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ such that
$$f(x,y)\lt 0\Leftrightarrow ...

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### Is a function with nowhere vanishing derivatives analytic?

My question is the following: Let $f\in C^\infty(a,b)$, such that $f^{(n)}(x)\ne 0$, for every $n\in\mathbb N$, and every $x\in (a,b)$. Does that imply that $f$ is real analytic?
EDIT. According to a ...

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### If $f$ is $C^{\infty}$ and $f^2$ is analytic, then $f$ is analytic

Assume that $f:\mathbb{C}^n\rightarrow \mathbb{C}$ is a $C^{\infty}$ function such that $f^2$ is (complex) analytic. Then one can show that $f$ is analytic. (Note: Liviu Nicolaescu and Alexandre ...

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### Surjective entire functions without critical points

It is easy to construct surjective locally univalent holomorphic functions $f: {\mathbb D}\to {\mathbb C}$, where ${\mathbb D}$ is the open unit disk.
I am pretty sure that the answer to the ...

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### Analytic perturbation of the eigenvalues/eigenvectors of non-Hermitian matrix

Consider a matrix function $A(x)$, analytically depending on single parameter $x$.
Consider the eigenvalue/eigenvector pair of $A(0)$, namely $\lambda(0)$ and $w(0)$.
The question is whether we can ...

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1k views

### An extension of Morera's Theorem

Morera's Theorem states that
If $f$ is continuous in a region $D$ and satisfies $\oint_{\gamma} f = 0$ for
any closed curve $\gamma$ in $D$, then $f$ is analytic in $D$.
I have two questions:
...

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### If $f(x)+f(2x)$ is quasianalytic, is $f(x)$ necessarily quasianalytic?

Assume that $f\in C^{\infty}$ and that $M_n$ is a sequence such that $$\sum_{n=0}^{\infty}\frac{M_n}{(n+1)M_{n+1}}=\infty$$
and for certain compact neighborhood of the origin $U$ of $\mathbb{R}$, ...

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173 views

### Inequality for certain analytic functions

Suppose that $f(z)$ is analytic for $\Re(z)>0$ and it is positive on the real axis. Suppose that for some $y_0$ we have
$$
\left| {f\left( {x + iy} \right)} \right| \le f\left( x \right)
$$
for any ...

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143 views

### analytic solution to elliptic PDE in R^n

I am looking for (minimal) conditions, which guarantee that the problem
Lu = 0 in R^n,
where L is a second-order (uniformly) elliptic operator with analytic coefficients, has a unique global ...

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569 views

### construct a power series with infinitely many zeros in the complex plane, bounded coefficients???

Hi all.
I want to construct a power series $F(z)=\sum_{n=0}^\infty c_nz^n$ centered at zero and with finite radius of convergence in the complex plane, and which has infinitely many zeros (in its ...

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### Univalent functions with non-negative coefficients

Is anything non-trivial known about univalent functions with non-negative coefficients?
Let $U$ be the unit disc, and $f$ a univalent (=injective) holomorphic function, $f(0)=0$,
$f'(0)>0$. ...

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### Does Physics need non-analytic smooth functions?

Observing the behaviour of a few physicists "in nature", I had the impression that among the mathematical tools they use a lot (along with possibly much more sofisticated maths, of course), there is ...

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### Analyticity of the solutions of PDE

Let's consider a (partial) differential equation with analytic coefficients. The initial conditions may be non-analytic*.
Is it possible a solution $f$, which is non-analytic at any point of the ...

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### Does a bounded real function have an analytic continuation [closed]

Consider the function $f:[0,1]\rightarrow\mathbb{R}$, where
$f$ is real-analytic on the open interval $(0,1)$
$f$ is bounded on the closed interval $[0,1]$ (ie. there is some constant $C$ such that ...

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### Analytic implicit function theorem

I'm looking for a proof of the analytic implicit function theorem (IFT). The only related proof I could find was the holomorphic inverse function theorem (by Henri Cartan). On Wikipedia, the analytic ...

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### Integral kernel of form $e^{-<x,y>^2}$

Let $K(x,y): \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ given by $K(x,y) = e^{-< x,y>^2}$ where $<\cdot,\cdot>$ denote the canonical inner product. Define integral operator ...