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

Is the set of polynomials in $H^{\infty}(\mathbb{D})$ weak-star dense in $H^{\infty}(\mathbb{D})$?
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### 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
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. ...
• 3,109
1 vote
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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• 3,109
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### 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 ...
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1 vote
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1 vote
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### 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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### 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 ...
• 1,691
1 vote
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
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### 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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### 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 ...
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 ...
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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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 \$\... 