## New answers tagged cv.complex-variables

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

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 ...

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}...

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 ...

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 ...

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 $...

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 RouchÃ©'s theorem or argument principle:
choose a very small arc $a1b$ of the unit circle so that $a$ is above the real line, and $b$ ...

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) = \...

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
$$
f(x)=\sum_{j=0}^\infty a_j\cos(2\pi jx),
$$
...

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&...

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')...

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-\...

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,...

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:
$$ \phi = f \mapsto \sup_{z \in \mathbb{D}} |f(z) - |z|| $$
...

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 &...

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,...

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) \...

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