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5 votes

Zeroes of entire function on $\mathbb C^n$

This follows immediatelly from the following paper: The Zero Set of a Real Analytic Function
Nick S's user avatar
  • 1,990
9 votes
Accepted

Conjectured closed form of $\int_0^1 \frac{\text{Li}_2\left(\frac{x}{4}\right)}{4-x}\,\log\left(\frac{1+\sqrt{1-x}}{1-\sqrt{1-x}}\right)\,dx$

As Henri Cohen remarked, the identity to prove is equivalent to $$\sum_{n=1}^\infty \frac{H_n^{(2)}}{(n+1)(2n+1)\binom{2n}{n}}=\frac{\pi^4}{972}.\tag{1}$$ In turn, this follows readily from the OP's ...
GH from MO's user avatar
  • 96.9k
4 votes

Mellin-Barnes integral representation of the exponential function with a non-real argument

Might be helpful to accentuate the poles by expressing the M-B contour integral in more standard notation as $$ e^{-p \cdot x} = \frac{1}{2\pi i} \int_{\sigma - i \infty}^{\sigma + i \infty} \frac{\pi}...
Tom Copeland's user avatar
  • 9,817
1 vote

Problem in understanding maximum principle for subharmonic functions

Proposition $:$ If $f$ is a u.s.c. function on $\Omega$ and bounded above, then there is a sequence $f_1 \geq f_2 \geq \cdots$ of continuous functions on $\Omega$ that are bounded above and that ...
Arnab Chattopadhyay.'s user avatar
0 votes

Accessible points of a simply connected domain

Nice references are Section 17 of Milnor's book "Dynamics in One Complex Variable" (old version is available in arXiv https://arxiv.org/abs/math/9201272 and it is Section 15 for this ...
inyo's user avatar
  • 36
7 votes
Accepted

Upper bound for the $n$-th derivative of a rational function $\frac{f}{f+g}$

There are counterexamples. Indeed, assume that the claimed bound holds for $x=1$. Then the Taylor series of $h(z)$ around $z=1$ converges in the disk $D=\{z:|z-1|<1\}$, hence $h(z)$ is analytic in $...
GH from MO's user avatar
  • 96.9k
2 votes

Roots of $x^n-x^{n-1}-\cdots-x-1$

For $P(x)=(1-x)f(x)=(x^{n+1}+1)-2x^n$ you may apply a version of Rouché's theorem or argument principle: choose a very small arc $a1b$ of the unit circle so that $a$ is above the real line, and $b$ ...
Fedor Petrov's user avatar
1 vote

Infinite sum of even Bessel functions - Identities

It appears to me that these identities can be interpreted in terms of Neumann series. To briefly sketch the idea, suppose that $f(z)$ is analytic in some disc centered at the origin. Then $f(z) = \...
Matt Young's user avatar
  • 4,633
3 votes

Limit of an infinite series with quadratic arguments

We will use the following well known fact (two proofs of this fact can be found at the end of the post): Given $f(x)$ with period $1$, its Fourier series $$ f(x)=\sum_{j=0}^\infty a_j\cos(2\pi jx), $$ ...
Cave Johnson's user avatar
  • 5,282
5 votes
Accepted

Infinite sum of even Bessel functions - Identities

Let $L$ denote the left-hand side of your identity \eqref{2}. Then, using the identity $$J_a(x)=\sum_{m\ge0}\frac{(-1)^m}{m!(m+a)!}(x/2)^{2m+a} \tag{$\dagger$}\label{3},$$ we get $$ \begin{split} L&...
Iosif Pinelis's user avatar
1 vote
Accepted

Small phase approximation

Linearization in $\phi_k(r)$ gives the desired approximation: $$S=\left|\int_0^\infty dr\, A(r)e^{-i[\phi_0(r)+\sum_{k=1}^n \phi_k(r)]} dr \right|^2$$ $$=\int_0^\infty dr\,\int_0^\infty dr'\, A(r)A(r')...
Carlo Beenakker's user avatar
3 votes
Accepted

Characterizing the unimodular functions from the closed disk $\overline{\mathbb{D}}$ to $\mathbb{C}$ with constraints

A description of your class $S$ can be obtained as follows. Let $z_0=1/\sqrt{2}$. Let $H$ be the class of all functions of the form $$f(z)=ze^{it}\frac{z-z_0}{1-z_0z}\prod_{n=1}^N\frac{z-z_n}{1-\...
Alexandre Eremenko's user avatar
4 votes

Entire function of finite order with deficient value

The theorem you stated, and its various versions and generalizations, are the only simple sufficient conditions for $\delta(0)>0$. For example, if $f$ is entire of genus $1$, and zeros lie on a ray,...
Alexandre Eremenko's user avatar
0 votes

Best approximation of the modulus function

Let $\mathbb{D} \subset \mathbb{C}$ be the unit disc. Let $\phi : (\mathbb{D} \to \mathbb{C}) \to \mathbb{R}$ be the following functional: $$ \phi = f \mapsto \sup_{z \in \mathbb{D}} |f(z) - |z|| $$ ...
user76284's user avatar
  • 1,793
1 vote
Accepted

Necessary conditions for convergence of convolution

Actually, here's a very strong counterexample (to the idea that disjoint ROCs lead to divergent convolutions), riffing off of my earlier comment. Let $S = \{ (-2)^k : k \in \mathbb{N} \}$. Choose $0 &...
Willie Wong's user avatar
  • 36.5k
2 votes

Interpolation by holomorphic functions of small exponential type on a half-plane

This is not always possible under your conditions. For example, if $a_n=0$ for $n\geq 2$, then any function of exponential type $<\pi$ interpolating this sequence must be zero by Carlson's theorem,...
Alexandre Eremenko's user avatar
1 vote

The inverse of the digamma function

We want to solve for $x$ the equation $$\psi ^{(0)}(x)=a$$ Just four months before your question, this paper gave tight bounds $$x_{\text{min}}=\frac{1}{\log \left(1+e^{-x}\right)} \lt \psi^{-1} (x) \...
Claude Leibovici's user avatar

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