# Questions tagged [analytic-functions]

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93
questions

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47 views

### Holomorphic semigroups vs analytic semigroups

Is there any difference between the two notions in the theory of semigroups?
In the literature, we find some monographs use the farmer while others use the latter. I expect that they are always the ...

**2**

votes

**1**answer

275 views

### real analytic function with given shape

I am looking for a 5 parameter family of analytic functions $f:[0,1]\to R$ 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, $p$ ...

**1**

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**0**answers

37 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**

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**1**answer

143 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 ...

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26 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 ...

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45 views

### Periodicity implied by the condition on level sets $\sum_{x \in \phi^{-1}(b)} \frac{\text{e}^{i \theta(x)}}{|\phi'(x)|}=0 $

Let $\theta$ and $\phi$ two real $C^h$ functions on $[0,2\pi]$, $h\geq1$, satisfying for all not critical values $b$ of $\phi$
$$
\sum_{x \in \phi^{-1}(b)} \frac{\text{e}^{i \theta(x)}}{|\phi'(x)|}...

**1**

vote

**1**answer

186 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 $\{...

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**1**answer

175 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**

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**2**answers

297 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 $=\pi/2$. ...

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59 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)>...

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74 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

**1**answer

154 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

**1**answer

107 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**

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**1**answer

329 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 ...

**10**

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**0**answers

332 views

### On Riesz criteria for Riemann hypothesis:

While Reading the book "Equivalents of Riemann hypothesis" by Kevin Broughan I came across the Riesz criteria for Riemann hypothesis .
Riesz defined a function :
$R(x) = \sum_{n=1}^\infty \frac {(-1)...

**5**

votes

**2**answers

192 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**

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**0**answers

407 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 ...

**0**

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39 views

### Bloch space and holomorphic functions

Assume that $f$ is on Bloch space of the holomorphic functions on the unit disk: $\sup_{|z|<1} (1-|z|)^{1/2} |f'(z)|<\infty$ and assume the same for $g$ and assume that $h(z)=\sqrt{g^2(z)+g^2(z)}...

**6**

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213 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

**1**answer

81 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

**1**answer

234 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, ...

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765 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

**2**answers

155 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 ...

**0**

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**1**answer

170 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**

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574 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 ...

**4**

votes

**1**answer

221 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 $...

**10**

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799 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 (\...

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139 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**

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**0**answers

95 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 ...

**4**

votes

**2**answers

889 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

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175 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

**1**answer

96 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}_{...

**7**

votes

**1**answer

333 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**

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**2**answers

187 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 ...

**5**

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**2**answers

323 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**

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**1**answer

154 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

**1**answer

261 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 ...

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**1**answer

159 views

### Hartogs' theorem in Banach spaces

In complex analysis one learns Hartogs' theorem:
Let $U\subseteq \mathbb{C}^n$ open and $f: U \rightarrow \mathbb{C}$ a function. Then $f$ is analytic iff for all $1\leq i \leq n$
$$ z \mapsto f(...

**3**

votes

**1**answer

245 views

### Analytic solutions to algebraic differential equation

Dear Colleagues and Friends,
Here I need to find some good reference on a subject that seems very much studied: sorry, if the rest of this question is too naive.
I believe that it's known that if a ...

**3**

votes

**1**answer

274 views

### Identity Theorem for Real-Analytic Hypersurfaces

There's an interesting statement it seems I can prove, but I can't find any references for it, which makes me suspicious of it. So, could someone verify that the statement is correct/incorrect or ...

**5**

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**1**answer

383 views

### how to pass from algebraic power series to the analytic ones

Fix a field of zero characteristic, $k$, e.g. $\Bbb{R}$ or $\Bbb{C}$. Suppose $k$ is normed (and complete for its norm). Consider the ring extensions: $k[x_1,..,x_n]\subset \ k<x_1,..,x_n> \ \...

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votes

**1**answer

198 views

### Morrey & Grauert - real analytic vector bundles admits analytic Riemannian metric

In theorem 1.2 of Brian Conrad's handout Operations with Pseudo-Riemannian metrics, the author writes
Theorem 1.2. Every $C^p$ vector bundle $E\to M$ over a $C^p$ manifold with corners $0\leq p\leq ...

**3**

votes

**3**answers

579 views

### When does $f^{-1}=\frac{1}{f}$ with $f$ a function mapping $\mathbb{R}^{*}$ to $\mathbb{R}$?

In mathematics, an inverse function is a function that "reverses" another function: if the function $f$ applied to an input $x$ gives a result of $y$, then applying its inverse function $g$ to $y$ ...

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**1**answer

271 views

### Uniqueness theorem for conformal mapping

Let $f$ and $g$ be analytic functions in the unit disk $D$, continuous in the closed disk and locally univalent, $f'(z)\neq 0,\; g'(z)\neq 0,\; z\in D$.
Assume that each has only finitely many ...

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vote

**1**answer

127 views

### Generalisation of Chebyshev series to arbitrary sets

A Lipschitz continuous function $f : [-1,1] \to \mathbb{C}$ has a unique representation as a series in terms of the Chebyshev polynomials $T_k$,
$$
f(x) = \sum_{k = 0}^\infty a_k \, T_k(x)
\qquad
\...

**18**

votes

**1**answer

1k views

### When do real analytic functions form a coherent sheaf?

It is known that, in general, the sheaf of real analytic functions on a real analytic manifold is not coherent. However, there are some examples, where we have coherence: for example, if $X$ is a ...

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18 views

### spiral forward orbits of analytic functions near repelling fixed points

An anonymous referee informs me that forward orbits near fixed points of analytic functions, such that the members of the forward orbits lie on spirals, are well-known. His citation for this (p. 31 of ...

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**1**answer

534 views

### Can an analytic function defined on a maximal torus be extended analytically to all the Lie group?

Let $G$ be a compact group and $T$ a maximal torus on $G$. Suppose $f$ is an analytic function defined on $T$. Is there an analytic function $F$ on $G$ whose restriction agrees with $f$ on $T$?

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66 views

### Are real polynomial maps regular in the sense of Gabrielov?

Let $\varphi: X\to Y$ an real analytic map of real analytic manifolds
and $x\in X$. Let us write $\mathcal O_{X,x}$ for the algebra of germs at $x$ of real analytic functions on $X$ and $ \widehat{\...

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177 views

### Semi-algebraicness of cells involved in integrals of semi-algebraic functions

Background: In "Stability under integration of sums of products of real globally subanalytic functions and their logarithms", by R. Cluckers and D.J. Miller, it is shown that the integral of a ...