Questions tagged [analytic-functions]

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2
votes
0answers
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
1answer
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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0answers
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
votes
1answer
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 ...
1
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0answers
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 ...
0
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0answers
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
1answer
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 $\{...
-3
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1answer
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
votes
2answers
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 $=\pi/2$. ...
1
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0answers
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)>...
3
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0answers
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
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1answer
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
1answer
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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1answer
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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0answers
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
2answers
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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0answers
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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0answers
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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0answers
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
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1answer
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
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1answer
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, ...
17
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2answers
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
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2answers
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
votes
1answer
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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2answers
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
1answer
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
votes
1answer
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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0answers
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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0answers
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
2answers
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
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0answers
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
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1answer
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
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1answer
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
vote
2answers
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
votes
2answers
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
votes
1answer
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
1answer
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 ...
4
votes
1answer
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
1answer
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
1answer
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
votes
1answer
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> \ \...
4
votes
1answer
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
3answers
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 $y$ ...
8
votes
1answer
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 ...
1
vote
1answer
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
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1answer
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 ...
1
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0answers
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 ...
16
votes
1answer
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 $T$?
3
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0answers
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{\...
4
votes
0answers
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 ...