2
votes
1answer
74 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
1answer
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
0answers
132 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
1answer
149 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
1answer
228 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
1answer
114 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
1answer
188 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
2answers
317 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
2answers
600 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
2answers
404 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
0answers
315 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
0answers
202 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
2answers
705 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
1answer
391 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
3answers
855 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
4answers
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 ...
-2
votes
1answer
270 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
0answers
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
3answers
688 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
0answers
709 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 ...
4
votes
1answer
386 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
3answers
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
4answers
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
6answers
844 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
3answers
527 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
0answers
193 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
3answers
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
3answers
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
2answers
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
0answers
241 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
3answers
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
1answer
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
2answers
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
1answer
750 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
2answers
583 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
1answer
710 views

Current status of Bloch Constant and Landau Constant bounds

The Bloch constant B (based on a theorem introduced by André 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 ...
9
votes
1answer
863 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
1answer
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
1answer
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
1answer
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
1answer
601 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
1answer
848 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
3answers
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
3answers
511 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
9answers
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?