Questions tagged [analytic-functions]

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Conditions for an improper integral of a real-analytic function to be real-analytic

Let $U$ be a complex domain and suppose $f:U \times \mathbb{R} \rightarrow \mathbb{R}$ is continuous and, for all $t$, is real-analytic in $s$. That is, $f(s, t)$ is continuous and $g_t(s):=f(s, t)$ ...
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2 votes
0 answers
81 views

Existence of analytic function in disk algebra [closed]

Does there exist an analytic function $f\in A(\mathbb{D})$, where $A(\mathbb{D})$ is the disk algebra, such that $f(0)=0$ and the real part of $f(z)$ is strictly positive?
0 votes
0 answers
29 views

Maximal ideal space of $H^{\infty}(\mathbb{D})$

Is the maximal ideal space of $H^{\infty}(\mathbb{D})$ contained in the predual of $H^{\infty}(\mathbb{D})$?
0 votes
0 answers
41 views

Density of polynomials in $H^{\infty}(\mathbb{D})$

Is the set of polynomials in $H^{\infty}(\mathbb{D})$ weak-star dense in $H^{\infty}(\mathbb{D})$?
0 votes
0 answers
56 views

Predual of $H^{\infty}(\mathbb{D})$

Is the predual of $H^{\infty}(\mathbb{D})$ contained in the maximal ideal space of $H^{\infty}(\mathbb{D})$?
1 vote
0 answers
65 views

A basic question on analytic wave front set

Suppose $u$ is a smooth function on the closed unit disk centered at the origin in $\mathbb R^2$. Let us denote by $e_1$ and $e_2$ the unit vectors in the direction of $x$ and $y$ axis respectively. ...
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1 vote
0 answers
79 views

Analyticity of solutions to Schrödinger's equation

Take Schrödinger's equation on $\mathbb{R}$, $i\partial_t\psi(x,t)=H\psi(x,t)$. Assume that $\psi(x,0)$ has compact support. Using known integral formulas for the propagators, it is fairly ...
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0 votes
1 answer
81 views

Existence of an eigenpair for d-bar operator in the unit disck

Let $\overline{\partial}=\frac{1}{2}(\partial_{x}+\textrm{i} \,\partial_y)$ and let $D$ be the unit disc in the complex plane. For each $\lambda \in \mathbb C$, consider the problem: $$ \overline{\...
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4 votes
0 answers
81 views

Mellin transform of the Bessel function $Y_n$ of order $n \geq 2$

The Mellin transform of the function $h$, locally integrable on $(0,\infty)$, is defined by $$M[h,z] = \int_0^\infty t^{z-1} h(t) dt \tag{1}$$ For some functions $h$ the above integral is not ...
5 votes
0 answers
104 views

Is the space of analytic sections of a vector bundle a Fréchet space?

Let $\pi : E \to M$ a smooth vector bundle of finite rank, where both $E$ and $M$ are finite dimensional smooth manifolds, and $M$ is compact. I already know that the space $\Gamma^\infty(M,E)$ of ...
1 vote
0 answers
43 views

When does an analytic submanifold descend to the quotient?

Let $\pi : M \to B$ be a smooth principal bundle with group $G$, where $M$ is an analytic (Fréchet, in my case) manifold, $B$ is a smooth (Fréchet) manifold and $G$ is a smooth (Fréchet) Lie group. ...
13 votes
2 answers
430 views

Behavior of $|f'(z)|/(1+|f(z)|^2)$ as $|z| \rightarrow \infty$?

Let $f(z)$ be an entire holomorphic function in $\mathbb{C}$, and consider the real-valued function $$g_f(z)=\frac{|f'(z)|}{1+|f(z)|^2}.$$ If $f(z)$ is a polynomial, then it is easy to prove that $\...
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5 votes
0 answers
214 views

Is the volume functional analytic in the space of embeddings? What about locally?

Let $(M^{n+1},g)$ be an analytic Riemannian manifold and let $\Sigma^n$ be a closed analytic manifold. Denote by $\operatorname{Emb}(\Sigma, M)$ the space of all smooth (or maybe analytic) two-sided ...
3 votes
0 answers
60 views

Prescribing variations that preserve the area

Let $(M^3,g)$ be a Riemannian manifold and let $\varphi : \Sigma \to M$ be a two-sided embedding of a closed surface into $M$, with a unit normal $N$. Suppose that $\varphi$ is a regular point of the ...
0 votes
1 answer
62 views

Measure of preimage of Jordan disk under entire map

Let $f\colon\mathbb{C} \to \mathbb{C}$ be an entire map. For simplicity assume that $f$ is of finite type, i.e., it has finite set $S(f)$ of singular values. $S(f) \subset \mathbb{C}$ is a minimal (...
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4 votes
1 answer
360 views

Bounded real analytic function with bounded derivative and its higher order derivatives

Let $f : \mathbb{R} \rightarrow \mathbb{C}$ be a bounded analytic function such that its derivative is also bounded. What kind of bound can we get on the higher order derivatives of $f$? Does it ...
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10 votes
1 answer
286 views

On a variant of Carlson’s theorem

My question is on whether or not there exists some monotone strictly decreasing sequence of positive numbers $c_1>c_2>\ldots$ such that given any $f$ which is a uniformly bounded holomorphic ...
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3 votes
1 answer
133 views

Exponentials and other functions of sums of anti-commuting operators

I know that if $A$ and $B$ are commuting operators, then $\exp(A+B) = \exp(A) \exp(B)$. Is there a similar formula if $A$ and $B$ are anti-commuting (that is, $AB+BA = 0$)? I have developed a formula ...
8 votes
2 answers
363 views

Literature on non-Archimedean analogues of basic complex analysis results

It looks like there is some literature out there on what might be called 'non-Archimedean complex analysis' e.g. Benedetto - An Ahlfors Islands Theorem for non-archimedean meromorphic functions and ...
5 votes
1 answer
186 views

When do volumes depend real-analytically on the parameters defining the regions?

Suppose $f_1, \dots, f_n$ are real-valued real analytic functions defined on an open set $B=(0,1)^d$ of $\mathbb{R}^d$. For $r \in \mathbb{R}$, let $S_r$ be the sub-level set in $B$ defined by the ...
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11 votes
1 answer
503 views

An extension of the Carlson's theorem in complex analysis

For the statement of Carlson's theorem please see, https://en.wikipedia.org/wiki/Carlson%27s_theorem. There is an extension of Carlson's theorem that says that the condition that $f$ needs to vanish ...
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4 votes
1 answer
562 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>...
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6 votes
1 answer
558 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+\...
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4 votes
2 answers
555 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
0 answers
100 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 $$\...
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6 votes
1 answer
273 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 \...
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2 votes
0 answers
89 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
0 answers
97 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, ...
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1 vote
2 answers
264 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 ...
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1 vote
0 answers
142 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 $$...
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3 votes
1 answer
211 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
1 answer
316 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 ...
3 votes
1 answer
67 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)...
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1 vote
0 answers
80 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 ...
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6 votes
1 answer
350 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 $\...
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14 votes
0 answers
670 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
3 answers
529 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
0 answers
41 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\...
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4 votes
1 answer
212 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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1 vote
0 answers
32 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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1 vote
1 answer
224 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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-3 votes
1 answer
192 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 ...
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4 votes
2 answers
330 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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1 vote
0 answers
74 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
0 answers
83 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(...
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0 votes
1 answer
319 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 ...
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2 votes
1 answer
237 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
1 answer
911 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
0 answers
473 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 ...
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5 votes
2 answers
238 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 $\...
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