# Tagged Questions

**2**

votes

**1**answer

65 views

### M-Wright function asymptotics

Let $M(z;\nu):= \frac{1}{\pi}\sum_{n=1}^{\infty} \frac{(-z)^{n-1}}{(n-1)!}\Gamma(\nu n)\sin(\nu n\pi)=\frac{1}{2\pi i}\int_{\text{H}_a}\exp(\sigma -z\sigma^{\nu})/\sigma^{1-\nu} d\sigma$, ...

**1**

vote

**1**answer

112 views

### On the geometry of roots of a sum of complex linear fractions

I was wondering if there is anything known about the geometry (position) of the roots of a rational map of the form
$$R(z):= \sum_{j=1}^{n} \frac{a_j}{z-p_j},$$
where the $a_j$'s are nonzero complex ...

**4**

votes

**0**answers

126 views

### semiclassical proof of Wigner semicircle

In Terence Tao's discussion of the Gaussian Unitary Ensemble, he derives the Dyson and Airy kernels. The GUE is the probability distribution of the eigenvalues of a random Hermitian matrix.
\[ \int ...

**2**

votes

**1**answer

147 views

### Interpolating delta like functions by trigonometric polynomials of bounded modulus and fast decay

Consider a grid of points $T=\{t_0,t_1,\ldots,t_m\}$ with $0\le t_i\le 1$. I would like to find a function $f(t):[0,1]\rightarrow \mathbb{C}$ of the form
\begin{equation*}
f(t)=\sum_{k=-n}^n c_k ...

**1**

vote

**1**answer

225 views

### A counterexample to the Polya-Schur master theorems for half-planes

Given an integer $n\ge 1$ we say that $f\in C[z_1,\ldots,z_n]$ is stable if $f(z_1,\ldots,z_n)\neq 0$ whenever $\text{Im}\ z_i>0$ for all $1\leq i\leq n$.
Stable polynomials with all real ...

**2**

votes

**1**answer

113 views

### Entire functions of exponential type with small $L^1$ norm outside a finite real interval

I'm interested in entire functions of exponential type $\sigma$ (Bernstein space $B_\sigma^1$) following
$$\int_{-\infty}^{\infty} |f(x)|dx=1,$$
whose norm is as small as possible outside a range ...

**0**

votes

**1**answer

186 views

### Asymptotic behavior of entire functions

Which entire function $f\left(x\right)$ goes asymptotically to $\dfrac{e^{-x}}{x}$ as $x$ goes to infinity with $x$ positive? That is, $\left(e^{-x}/x \right)/f \left(x \right) \rightarrow 1$.

**11**

votes

**2**answers

314 views

### Effective vanishing of the Schwarzian Derivative

Recall for any complex analytic function $f:\mathbb{D}\to \mathbb{C}$
the Schwarzian derivative of $f$ is
$$
S(f)=\frac{f'''}{f'}-\frac{3}{2} \left( \frac{f''}{f'}\right)^2.
$$
It's well known that ...

**9**

votes

**2**answers

595 views

### Complex evaluation of a classical (real) integral

There are several ways to compute the classical integral
$$
\int_{\mathbb R}e^{-x^2}dx=\sqrt{\pi}.
$$
Probably, best known are
(1) squaring the integral with subsequent change
of (now two) variables ...

**3**

votes

**2**answers

402 views

### Question on a Basel-like sum

Hello all,
I have happened upon the following sum:
$ 1^2 + \Big(1 \times \frac{1}{3} + \frac{1}{3} \times 1 \Big)^2 + \Big(1 \times \frac{1}{5} + \frac{1}{3} \times \frac{1}{3} + \frac{1}{5} \times ...

**2**

votes

**0**answers

309 views

### Gaussian type integral with inverse square root

Hi,
I have encountered an integral of the following type in an engineering application:
$\int_{-\infty}^\infty dx \frac{1}{\sqrt{x^2+a^2}}\exp(-x^2/2+i x b)$,
where $a$ and $b$ are real ($a$ could ...

**0**

votes

**0**answers

201 views

### is there a relation between the complex Hardy spaces and the Hardy spaces of harmonic analysis?

Maybe my question is just a matter of knowing the right equivalent definition.
The question is whether there is some relation between
$ H^p(D^2)$, defined as the space made of the analytic ...

**4**

votes

**2**answers

689 views

### General form of Schwarz reflection principle

Hello all,
It is easy to find results on reflecting holomorphic functions over circles and lines, but I am wondering what there is for reflecting over smooth curves, or analytic arcs, etc. In ...

**8**

votes

**1**answer

390 views

### Properties of a matrix-valued generalization of the $\Gamma$ function

I am interested in the following function (Mellin transform of matrix exponential):
$$\int_0^{\infty} x^{s-1} e^{-A-Bx}d x$$
Where $x$ and $s$ are scalars, but $A$ and $B$ are matrices with $B\succ ...

**2**

votes

**3**answers

845 views

### Show that holomorphic functions are infinitely differentiable without complex analysis [duplicate]

Possible Duplicate:
Is there an integration free proof (or heuristic) that once differentiable implies twice differentiable for complex functions?
Is there a way to show that holomorphic ...

**3**

votes

**4**answers

2k views

### Analytic implicit function theorem

I'm looking for a proof of the analytic implicit function theorem (IFT). The only related proof I could find was the holomorphic inverse function theorem (by Henri Cartan). On Wikipedia, the analytic ...

**-1**

votes

**1**answer

268 views

### holomorphic equation

hi,
i am working for some time on a problem and at some point i cant go further. here the critical part: Let $U \subset \mathbb{C}^{n}$ be a open set and consider $c : U \rightarrow \mathbb{R}$ a ...

**1**

vote

**0**answers

155 views

### Class of flat currents stable under $\overline{\partial}$ operator

Given $U\subset\mathbb{C}^n$, open domain, a locally flat current on $U$ is a $k-$current $T$ such that for every $f\in\mathcal{D}(U)$ (smooth functions with compact support in $U$) there exist a ...

**3**

votes

**3**answers

686 views

### Boundary behavior of a holomorphic function on $D$ ?

Hi, I have two related questions.
$D$ = open init disk in the complex plane $C$.
A. Let $f: D \to C $ be a holomorphic function. Then is it possible that $\forall q \in S^1$,there exists a ...

**5**

votes

**0**answers

708 views

### Is this Fourier integral well-known?

The following integral is a special case of one that arises in an economics problem:
$I(u_{1}, u_{2}) := \displaystyle \int_{z_{1}=-\infty}^{\infty} \int_{z_{2}=-\infty}^{\infty} \frac{ \displaystyle ...

**3**

votes

**1**answer

378 views

### Vandermonde-type identity for Jacobi theta functions?

My question concerns an application in physics. By Vandermonde identity I refer to the following statement: take $f_j (z)=z^j$, where $z=x+iy$ is a complex coordinate and $j$ an integer. Make an ...

**9**

votes

**3**answers

2k views

### Ramanujan's eccentric Integral formula

The wikipedia page on Srinivasa Ramanujan gives a very strange formula:
Ramanujan: If $0 < a < b + \frac{1}{2}$ then, $$\int\limits_{0}^{\infty} \frac{ 1 + x^{2}/(b+1)^{2}}{ 1 + ...

**11**

votes

**4**answers

1k views

### Routh-Hurwitz for eigenvalues

The Routh-Hurwitz criterion provides a convenient test, even for hand calculation, of whether a polynomial with real coefficients has all its roots in the left half plane. I'm wondering about a ...

**2**

votes

**6**answers

842 views

### Reference for complex analysis jargon

I am not a (complex) analyst but it seems that some of the questions I am working on are related to the following concepts:
logarithmic capacity
transfinite diameter
Green's function of a compact ...

**4**

votes

**3**answers

526 views

### Holomorphic function with a.e. vanishing radial boundary limits

Hello everybody.
I'm looking for an "easy" example of a (non-zero) holomorphic function $f$ with almost everywhere vanishing radial boundary limits: $\lim\limits_{r \rightarrow 1-} f(re^{i\phi})=0$.
...

**2**

votes

**0**answers

192 views

### Dimension of pluripolar sets

Let $\Omega$ be an open set in $\mathbb C^n$, and let $A$ be a closed pluripolar set in $\Omega$. Is there a notion of dimension of $A$ such that the following theorem is true?
Theorem.
Let $\phi$ ...

**5**

votes

**3**answers

1k views

### When I can safely assume that a function is a Laplace transform of other function?

If I have a function and I want to represent it as being the Laplace transform of another, that is, I want to be sure that there is $\hat{f}(s)$ such that my function $f(x)$ can be written as:
$f(x) ...

**5**

votes

**3**answers

1k views

### Product of sine

For which $n\in \mathbb{N}$, can we find (reps. find explicitly) $n+1$ integers $0 < k_1 < k_2 <\cdots < k_n < q<2^{2n}$
such that
$$\prod_{i=1}^{n} \sin\left(\frac{k_i \pi}{q} ...

**7**

votes

**2**answers

1k views

### The Cauchy-Riemann equations and analyticity

I would be interested to learn if the following generalization of the classical Looman-Menchoff theorem is true.
Assume that the function $f=u+iv$, defined on a domain $D\subset\mathbb{C}$, is ...

**3**

votes

**0**answers

240 views

### Transforming a multivariable integral to make it separable

In the following I will omit requirements of smoothness, extent of domain, finiteness, etc, both to simplify the exposition and because I don't know exactly what the requirements are. Please imagine ...

**34**

votes

**3**answers

4k views

### On linear independence of exponentials

Problem.
Let $\{\lambda_n\}_{n\in\mathbb N}$ be a sequence of complex numbers . Let's call a family of exponential functions $\{\exp (\lambda_n s)\}_{n\in\mathbb N}$ $F$-independent (where $F$ is ...

**11**

votes

**1**answer

1k views

### Let a function f have all moments zero. What conditions force f to be identically zero?

Throughout, let $f$ be a Lebesgue measurable function (or continuous if you wish, but this is probably no easier). (Questions with distributions etc. are possible also but I want to keep things simple ...

**27**

votes

**2**answers

2k views

### Must the set of lines through the origin on which a nonconstant entire function is bounded be finite?

If an entire function is bounded for all $z \in \mathbb{C}$, than it's a constant by Liouville's theorem. Of course an entire function can be bounded on lines through the origin $z=r \exp(i \phi), ...

**5**

votes

**1**answer

739 views

### Approximation by analytic functions

Dear all.
Let
$$
f(x) = \sum_{k \in \mathbb{Z}} \hat{f}(k) \exp(2\pi \mathrm{i} kx)
$$
be a function given by usual fourier series.
Since my original question hasn't got any answer yet, and I ...

**11**

votes

**2**answers

581 views

### Asymptotic approximation of $x^\alpha$ by entire functions

Given a non-integral real $\alpha$, is there an entire (see http://en.wikipedia.org/wiki/Entire_function) function $h(x)$ such that $x^{-\alpha}h(x)\longrightarrow 1$
for $x\rightarrow+\infty$ (with ...

**6**

votes

**1**answer

704 views

### Current status of Bloch Constant and Landau Constant bounds

The Bloch constant B (based on a theorem introduced by AndrĂ© Bloch in 1925 on the maximum radius of a one-to-one disk in the image of a normalized analytic function of the unit disk, see for instance ...

**8**

votes

**1**answer

859 views

### How to best distribute points on two concentric circles?

An N-subset $\{x_1,\dots,x_N\}$ of a compact set $X\subset \mathbb R^d$ is called a set of Fekete points (named after Michael Fekete) if it maximizes the product $$\prod_{1\le k<j\le ...

**0**

votes

**1**answer

240 views

### the Cech-cohomology of the sheaf of germs of plurisubharmonic functions defined on a domain in C^n

we all know that if we consider the sheaf of germs of a holomorphic functions defined on a domain in C^n,we have too many beautiful theorems characterizing the geometry of the domain by consider the ...

**3**

votes

**1**answer

385 views

### Is a function which is finitely multiple-valued in each variable separately, also finitely multiple-valued in all its variables jointly?

It is well known that under suitable conditions, a function which is:
a polynomial in each variable separately is a polynomial in all its variables jointly.
a rational function in each variable ...

**0**

votes

**1**answer

173 views

### An integral arising in statistics(2)

The integral I am interested in is:
$$t(x)=\int_{-K}^{K}\frac{\exp(ixy)}{1+y^{2q}}dy$$
$K<\infty$, q natural number
For q=1 one can use contour integration.
So for K>1 we have :
...

**9**

votes

**1**answer

595 views

### Approximation to divergent integral

Hi everyone,
I'm a physicist working on stochastic processes and I've come up against an integral that I'm not able to approximate using steepest descent (I don't have a large or small parameter), ...

**1**

vote

**1**answer

847 views

### How can I calculate the characteristic function of these distributions? [previously: difficult integral]

How to compute this integral in general case?
$$t(x)=\int_{-\infty}^{\infty}\frac{\exp(ixy)}{1+y^{2q}}dy$$
Mathematica can compute it when q is known. For example,for q=1 this integral is
...

**4**

votes

**3**answers

1k views

### Most important domains, extension theorems, and functions in several complex variables

For a new learner of several complex variables, the many domains (eg holomorphically convex, pseduconvex, Stein) and the many extension theorems (eg Riemann) and the many functions (plurisubharmonic) ...

**16**

votes

**3**answers

461 views

### When are some products of gamma functions algebraic numbers?

I want to know when certain expressions of the form
$ {\Gamma(r_1/m) \Gamma(r_2/m) \ldots \Gamma(r_j/m) \over \Gamma(s_1/m) \Gamma(s_2/m) \ldots \Gamma(s_j/m)} $
are algebraic numbers. These ...

**36**

votes

**9**answers

7k views

### Why do functions in complex analysis behave so well? (as opposed to functions in real analysis)

Complex analytic functions show rigid behavior while real-valued smooth functions are flexible. Why is this the case?