**5**

votes

**0**answers

89 views

### Complex L^1 spaces; reference request

I have been doing a fair amount of research into a complex analytic modified version of the Mellin transform. I have hit a few roadblocks, and am hoping there may already be literature on the subject. ...

**3**

votes

**1**answer

139 views

### Do some kind of maximum principle exist on complex manifold?

Consider a holomorphic function $u:C\to C^n$, as $|u|^2 $is sub-harmonic, it satifies a maximum principle.
Do some general kind of complex manifold enjoy such property? Say, square of some distance ...

**2**

votes

**0**answers

44 views

### Interpolation polynomial smaller than its function?

Let $q$ be a real number such that $q>1$ and $f$ be an entire function on $\mathbb C$ such that $\overline{\lim}_{r\to+\infty}\limits\frac{\ln|f|_r}{\ln^2r}<\frac{1}{2\ln q}$, where ...

**3**

votes

**0**answers

114 views

### Automorphism groups of elliptic bundles

This is a question in complex geometry, but even for algebraic varieties I don't know the answer:
Let $S$ be a smooth compact Kähler surface (for example a smooth complex projective surface) that is ...

**3**

votes

**1**answer

129 views

### Growth comparision between an entire function and a related function

Let $p$ be a prime number, $\mathbb C_p$ be the completion of an algebraic closure of $\mathbb F_q\left(\left(\frac1T\right)\right)$ for the valution $v(x)=-\deg(x)$. Let $\sum_{n\ge0}a_nz^n$ be a ...

**2**

votes

**1**answer

103 views

### The motivation and application of Nevanlinna second main theorem for small functions

I once read some books about Nevanlinna theory, most of them will discuss the Nevanlinna main theorem small function theorem under some conditions. While, I know little motivation of small function ...

**3**

votes

**0**answers

266 views

### Funk-Hecke theorem on the complex sphere

I am interested in paper " Sharp constants in several inequalities on the Heisenberg group " of Rupert L.Frank and Elliott H.Lieb " http://arxiv.org/pdf/1009.1410v2.pdf. In this paper ( page 17 ), ...

**0**

votes

**0**answers

44 views

### Explicit solution of a Cauchy-type singular integral equation with regular part

I am doing research on the Riemann boundary value problem for bi-half-planes, and in a certain case I was able to reduce this problem to a linear singular integral equation of the form
...

**0**

votes

**1**answer

71 views

### a sum of ratios of quadratic forms

I have the following function that I would like to optimize over the value A
$$f(A)=\sum_k \frac{\mathbf{y}_k^H\left[\begin{array}{cc} 1&0\\ 0& A \end{array} ...

**1**

vote

**1**answer

88 views

### Visualization of non-Smirnov domains

Can one provide a graph of a non-chord arc(non-Lavrentiev) Jordan curve in the plane? That is, more of less, equivalent to a Jordan curve whose interior domain is a non-Smirnov domain.

**3**

votes

**2**answers

172 views

### Sampling Theorem for non-bandlimited Functions

The classical Shannon sampling theorem states that
a bandlimited function with $\mbox{supp } \hat f\subset [-1/2,1/2]$ can be uniquely determined by its samples $(f(i))_{i\in \mathbb{Z}}$ (The symbol ...

**5**

votes

**1**answer

309 views

### About the logarithmic derivative of the Riemann zeta function

Let $\rho=\beta+i\gamma$ a non-trivial zeros of the Riemann zeta function and $s=\sigma+it$ a complex number. It is possible to prove that ...

**1**

vote

**0**answers

62 views

### summability and analytic continuation

Let $d_n=LCM(1,\cdots,n)$. It is well-known that $d_n=e^{\Psi(n)}$ where $\Psi$ est the second Chebyshev function. One knows that $\Psi(x)=\sum_{k\le x}\Lambda(k)$ where $\Lambda$ is the Von Mangold ...

**18**

votes

**3**answers

471 views

### Does $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converge to zero when $\alpha$ is irrational?

I came across a problem concerning about the convergence of products. I wonder if the complex series $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converges to zero when $\alpha$ is irrational. Of course, ...

**1**

vote

**1**answer

90 views

### On the horizontal behaviour of certain complex functions

Assume that $f(z)$ is holomorphic in $\{z=x+iy:\ 0\leq x\leq 1,\ t>2\}$ and it satisfies $|f(x +it)| = O(t^{(1-x)/2}\log t)$ for $0 \le x \le1$. In particular, $|f(it)| = O(t^{1/2}\log t)$. Is it ...

**0**

votes

**0**answers

111 views

### Extension of harmonic function with bounded $L^{2}$ norm

Let $h:D\setminus \{0\}\rightarrow \mathbb{R}$ be a harmonic function, where $D$ is the unit disc in $\mathbb{R}^{2}$, with bounded $L^{2}$ norm, i.e. $||h||_{L^{2}(D)}^{2}=\int_{D}|h|^{2}(x,y)dxdy ...

**1**

vote

**0**answers

69 views

### Expansion of a power series as integral of cosine functions

Suppose $$f(x)=\sum_{k=0}^\infty (-1)^k x^{2k} \int_0^1 p(\xi_{2k})\int_0^{\xi_{2k}}q(\xi_{2k-1})\cdots\int_0^{\xi_3}p(\xi_2)\int_0^{\xi_2}q(\xi_1) d\xi_1 \;d\xi_2\cdots d\xi_{2k-1}\;d\xi_{2k},$$
...

**4**

votes

**0**answers

132 views

### Contour integral $\int_{\gamma}e^{-t\mu}(-\mu)^{-m}\log\sqrt{-\mu}d\mu$

Motivation: In my research I try to find the asymptotics of the heat trace $\text{tr}e^{-t\Delta}$ as $t\to0$, where $\Delta$ is Laplace operator on a manifold with singularities. First I find the ...

**6**

votes

**3**answers

220 views

### On what kind of condition of a compact set $K$ in the plane, $C(K)$ has a generator?

Let $K\subset \Bbb{C}$ be a compact subset of the complex plane, and let $C(K)$ be the space of all complex continuous functions on $K$.
We say that $f\in C(K)$ is a generator of $C(K)$ when the set ...

**3**

votes

**1**answer

207 views

### bounded analytic function as a power series

Suppose $$f(x)=\sum_{k=0}^\infty a_k\frac{(i x)^k}{k!}$$ where $$a_k=k!\int_0^1 p_k(y_{k-1})\int_0^{y_{k-1}}p_{k-1}(y_{k-2})\cdots \int_0^{y_1}p_1(y_0) dy_0\cdots dy_{k-2}\;dy_{k-1}$$ for functions ...

**3**

votes

**2**answers

160 views

### Lower Bounds for the Roots of Polynomials

I'm interested in the "size" of the roots of a sequence of Taylor Polynomials of an entire function.
For example, consider $\mathrm f(z) = \mathrm e^z$. The Taylor Polynomials, or $k$-jets, are
...

**4**

votes

**1**answer

194 views

### Artin approximation vs implicit function theorem in the class of analytic functions

I asked this on math stackexchange but I had no luck, so I am posting my question also here.
I am not an algebraist so my question might be stupid. I am doing mainly complex analysis and recently I ...

**3**

votes

**2**answers

176 views

### Bergman norm on a bigger domain

Let $D$ be a unit disc (I am actually interested in a much more general setting, but let's start with explicit examples). Let $E$ be an open subset of $D$. Consider the functional on the space of all ...

**25**

votes

**0**answers

444 views

### “Three great cocycles” in Complex Analysis as cohomology generators

In his lecture notes, C. McMullen discusses "the three great cocycles" in Complex Analysis: the derivative $$f\mapsto\log f',$$ the non-linearity $$f\mapsto (\log f')'dz$$
and the Schwarzian ...

**4**

votes

**2**answers

93 views

### Is there a critical point of a polynomial $f$ within every disc having as diameter the line segment between two zeros of $f$?

Consider a generic complex polynomial $f$ (i.e. one with pairwise distinct zeros [EDITed:] and with pairwise distinct zeros of $f'$) and define for every pair of zeros an open disc which has as ...

**3**

votes

**0**answers

159 views

### Combination of Generating Functions

Suppose I have the following generating functions:
$$\frac{x^ke^{\left(z-\frac{1}{N}\right)x}}{N^{k-1}k!\sum_{j=0}^{N-1}w_N^{-jk}e^{\frac{w_N^jx}{N}}}=\sum_{j=0}^\infty H_{N,k,j}(z)\frac{x^j}{j!}$$
...

**6**

votes

**1**answer

101 views

### Coefficient problem for univalent harmonic functions on unit disk

The Clunie Sheil Small conjecture for the second coefficient of a univalent harmonic function on the unit disk is as follows:
Suppose, $h(z)+\overline{g(z)}$ is a one-to-one harmonic function on the ...

**1**

vote

**0**answers

89 views

### approximation of rational functions

Suppose $\hat{p}/\hat{q}$ and $p/q$ are two rational functions where $p,q,\hat{p},\hat{q}$ are of degree $n$. Suppose they satisfy that $|p(z)/q(z) - \hat{p}(z)/\hat{q}(z)| < \epsilon$ for any $z$ ...

**4**

votes

**2**answers

113 views

### existence of a special conformal mapping

Sorry I don't know how to give an appropriate title.
In the complex plane, suppose there is a graph $x+if(x)$ separating the plane into two unbounded components, where $f(x)$ is smooth and bounded, ...

**2**

votes

**1**answer

168 views

### Strange limit problem involving $\binom{z}{n} e^{-xn\log n}$ with $z \in \mathbb{C}$

Is the following limit result correct: $$\lim\limits_{x \to 0^{+}}\sum\limits_{n=0}^{\infty} \binom{z}{n} e^{-xn\log n} = 2^{z}$$ where, $z \in \mathbb{C}$, and the notation $\displaystyle ...

**8**

votes

**4**answers

382 views

### Analytic continuation of the double sum $\sum_{n,m\ge0}x^ny^mt^{nm}$

Define function $f(x,y,t)$ as the analytic continuation of the series
$$f(x,y,t)=\sum_{n,m\ge0}x^ny^mt^{nm}$$
This series definitely converges when all the arguments are small enough. I would like to ...

**1**

vote

**1**answer

114 views

### When is a homogeneous polynomial an inner function on the unit torus?

What is the necessary and sufficient condition(s) for a homogeneous polynomial on torus(bidisk) to be an inner function? More precisely I want to know when absolute value of a homogeneous polynomial ...

**1**

vote

**1**answer

91 views

### A Gaussian integral over complex variables by a defined Green's function for a Gaussian ensemble of random matrix

We construct an $N\times N$ matrix $J$ whose elements are drawn from Gaussian distribution with zero mean and variance $\frac{1}{N}$. Since we want to have different variances for different columns, ...

**3**

votes

**1**answer

100 views

### What is the correct generalization of the Wirtinger derivatives to arbitrary Clifford algebras?

In the complex numbers setting, the two Wirtinger derivatives are defined as:
$\frac{\partial}{\partial z}=
\frac{1}{2}\left(
\frac{\partial}{\partial x} - i
\frac{\partial}{\partial y}
...

**0**

votes

**0**answers

64 views

### Existence of analytic continuation of $f(z)=\sum{n^{\alpha}} z^n$ for fractional $\alpha$

The power series
$f(z)=\sum_{n \ge1}{n^{\alpha}} \cdot z^n$
has radius of convergence 1. For $\alpha \in \mathbb{N}$ it is easy to see that $f$ permits an analytic continuation to $\mathbb{C} ...

**1**

vote

**2**answers

138 views

### Examples of pluripolar sets

I have a very basic question on pluripolar sets. First remind their definition.
Let $\Omega\subset \mathbb{C}^n$ be a domain. A subset $E\subset \Omega$ is called pluripolar if there exists a ...

**1**

vote

**0**answers

63 views

### Inverse Laplace transform of a non-negative function

Consider an entire function $f$, which is real for real arguments and satisfies $f(s)\geq 0$ for all $s\in\mathbb{R}$. Furthermore, assume this function is a Laplace transform,
$$
f(s)=\int_0^\infty ...

**1**

vote

**0**answers

94 views

### Does Feuter regularity imply derivability in all directions?

The standard type of regularity in Clifford Calculus is the one introduced by Feuter, namely:
a function is Feuter regular iff it is in the zero set of the Clifford-Dirac
operator $D= ...

**30**

votes

**3**answers

1k views

### Polynomials with the same values set on the unit circle

Assume that $P(z)$, $Q(z)$ are complex polynomials such that $P(S)=Q(S)$, where $S=\{z\colon |z|=1\}$ (equality is understood in the sense of sets, but I do not know the answer even for multisets). ...

**3**

votes

**0**answers

206 views

### Inverse problem for negative moments

Let $D$ be a bounded simply connected domain in the plane, bounded by a smooth closed curve $\partial D$. Moreover$D$ contains the origin. Assume that all negative complex moments w.r.t arc-length ...

**2**

votes

**0**answers

53 views

### Holomorphic solution to SDE

Consider the SDE $dZ_t = \mu(t,x) d_t + \sigma(t,x) dW_t$.
Are there any known (necessary and) sufficient conditions on $\sigma(t,x)$ and on $\mu(t,x)$ guaranteeing that $f(T):=\mathbb{E}[\int_0^T Z_t ...

**9**

votes

**3**answers

429 views

### Complex proof of $B(a,b)=\Gamma(a)\Gamma(b)/\Gamma(a+b)$

It is a question in spirit of this one.
Is there a way to prove Euler's formula
$$
\int_0^1 x^{a-1}(1-x)^{b-1}dx=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}
$$
using contour integration (and maybe ...

**0**

votes

**0**answers

47 views

### How to take partial derivative of spherical interpolation of quaternions?

Using the standard definition of quaternionic spherical linear interpolation (slerp):
$$
Q(q_0,q_1,t) := q_0(q_0^{-1}q_1)^t,
$$
how can I take each partial derivative?
Actually, I'm confident how to ...

**1**

vote

**0**answers

93 views

### For which integer values of $k$ can we find one solution to the equation $\sum\limits_{n=1}^{n=k} \frac{1}{n^s}=0$ by iteration? [closed]

I am trying to find solutions to the well known equation:
$$\sum\limits_{n=1}^{n=k} \frac{1}{n^s}=0$$
Now with this program below I have found that for certain values of the integer $k$ one can find ...

**0**

votes

**0**answers

70 views

### A question on evaluation of complex integrals

Is any general relationship between the integral
\[
\int_{0}^{1}f(u, \sigma + it)u^{-1 + d}du
\]
and $f(0, \sigma + it)$ known?
I have proved one such result where the main term of the given
integral ...

**2**

votes

**0**answers

95 views

### system of complex number equations

Let $a_1,a_2,a_3,a_4\in \mathbb{C}$ be distinct such that
$$a_1^3+a_2^3+a_3^3+a_4^3=0$$
$$(1+|a_1|^2)a_1^2+(1+|a_2|^2)a_2^2+(1+|a_3|^2)a_3^2+(1+|a_4|^2)a_4^2=0$$
...

**6**

votes

**1**answer

149 views

### Can the potential of a complete Kahler metric be bounded?

Let $X$ be a complex manifold and $\omega$ a Kahler form on $X$. A smooth function $\rho$ is called a potential of $\omega$ if $i\partial\bar\partial\rho=\omega$. By intuition, it seems that $\rho$ ...

**-2**

votes

**1**answer

64 views

### Behavior of “integer complex number” on computer [closed]

I want to provide software to compute with "integer complex numbers", that live in $\mathbb{Z}\times i \mathbb{Z}$, rather than the $\mathbb{C}$. Some operations are going to give results that are ...

**4**

votes

**0**answers

71 views

### sums of zero-free entire functions and its siblings on the disk

Can one describe the set $\{e^f+e^g: f, g\in H(C)\}$ in some way?
For example, in unital Banach algebras, every element has this form.
I am in particular interested in the problem whether the ...

**2**

votes

**1**answer

76 views

### Octahedron and System of trigonometric equations

Could somebody help me to prove the following?
$$\sum_{k=1}^6 \cos(2 \theta_k) (\cos(2\phi_k)-1))=0$$
$$\sum_{k=1}^6 \sin(2 \theta_k) (\cos(2\phi_k)-1))=0$$
$$\sum_{k=1}^6 \cos (\phi_k)=0$$
...