**0**

votes

**0**answers

43 views

### How to get Wiener-Hopf decomposition of this matrix function

I met a matrix function $M(z)$ that I need to get its Wiener-Hopf decomposition that $M(z) = M_+(z) M_-(z)$ with $M_+(z)$ and $M_-(z)$ being analytic inside and outside the unit circle respectively. ...

**1**

vote

**2**answers

82 views

### Nonlinear PDE for a 2D foliation

I am trying to solve a 4th order nonlinear PDE for a real function $u(x,y)$ of two variables. It is too complicated to reproduce here but it exhibits the following two very nice properties:
1) if $u(...

**1**

vote

**0**answers

43 views

### Modulus of Continuity for an Analytic Function on an Ellipse

This is a question which I stumbled upon while working on Legendre Polynomials, but it is actually a question in complex analysis.
Consider: Given $f\in C^{\infty} (E)$, where $E_{\rho} \subseteq \...

**4**

votes

**1**answer

170 views

### Embed a bordered Riemann surface into punctured Riemann surfaces?

Let $U$ be a bordered Riemann surface of genus $g$ with $n -1$ punctures and one hole (i.e., the border has one connected component). For any punctured Riemann surface $\Sigma$ of genus $g$ with $n$ ...

**3**

votes

**1**answer

97 views

### Wiener-Hopf factorization of matrices

Given a $2\times2$ matrix, which entries are functions in the complex plane $$\hat{A}(z)=\left(\begin{array}{cc}a(z)&b(z)\\c(z)&d(z)\end{array}\right)$$Where $a(z),b(z),c(z)$ and $d(z)$ are ...

**2**

votes

**0**answers

113 views

### The boundedness of an entire function along the imaginary axis

I am looking for an answer in the affirmative or the negative concerning the asymptotics of an entire function. The question, relatively simple to state, has proven very taxing to solve and requires ...

**7**

votes

**0**answers

163 views

### Biholomorphic neighborhoods of the boundary of Stein domains

Let $(X_1,J_1)$ and $(X_2,J_2)$ be Stein domains with the same contact boundary $(Y,\xi)$. Under what conditions does there exist a biholomorphism between a neighborhood of their respective boundaries ...

**3**

votes

**1**answer

93 views

### $L^p$ norm of boundary values of holomorphic function

I am looking for an estimate of the following form:
Suppose that $D\subset \mathbb{C}$ is a simply connected domain. Suppose that $F$ is holomorphic and bounded on $D$ and can be holomorphically ...

**1**

vote

**0**answers

63 views

### Explicit formula of biholomorphism between the rectangle and unit disk [closed]

From the Riemann mapping theorem we know that there exists a biholomorphism between the rectangle $R$ and the unit disk $D$, can we write this biholomorphic map explicitly?

**0**

votes

**1**answer

134 views

### How to calculate the expected value of complex-valued random variable? [closed]

Suppose $\theta_1,\theta_2,\cdots, \theta_n$ are independent and identically distributed (i.i.d.) real-valued random variables and here we specifically consider the uniform distribution in the ...

**3**

votes

**0**answers

105 views

### Siegel's article “The volume of the fundamental domain for some infinite groups”: trouble with understanding computations

This is the article I mentioned. While the idea of what Siegel is doing in order to compute the volume of the fundamental domain described in the article (the very first one, for there are discussed ...

**6**

votes

**1**answer

116 views

### Convex Hull of univalent functions and Bieberbach Conjecture

Consider the class $S$ of univalent functions on the unit disk $D$ normalized so that $f(0)=0$ and $f'(0)=1$.
Each function in $S$ satisfy the Bieberbach conjecture, that is the $n$-th coefficient in ...

**-4**

votes

**1**answer

282 views

### Proof of formula for $\pi$ [closed]

The number $\pi$ can be expressed as $\pi=\lim_{n\to\infty} \frac{n\sqrt[n]{-1}-n}{\sqrt{-1}}$ or more poetically $\pi=\frac{\infty\sqrt[\infty]{-1}-\infty}{\sqrt{-1}}$. Here we choose the principal ...

**3**

votes

**0**answers

86 views

### State of the art for univariate complex polynomials factorization with algebraic coefficients

Let $\mathbb{K}:=\overline{\mathbb{Q}}$ be the field of algebraic numbers. We choose to represent an element of $\mathbb{K}$ as its minimal monic polynomial, which is a vector in some $\mathbb{Q}^n$.
...

**6**

votes

**2**answers

171 views

### Criterion for deciding the conformal class of a metric on a complete surface

For orientable closed Riemannian surfaces $(S,g)$, there is a constant curvature metric $\overline{g}$ on $S$ that is conformal to $g$ in the sense that $\overline{g} = e^ug$ for some smooth function $...

**2**

votes

**0**answers

102 views

### Asymptotic analysis of generating functions

Let $a_d\!\in\!{\mathbb R}^+$ with $d\!\in\!{\mathbb Z^+}$ be a sequence such that
$$\limsup \sqrt[d]{a_d}=1\,.$$
Define
$$F(z)=\sum_{d=1}^{\infty}a_d\,{\text{e}}^{d z}\,.$$
Suppose $F(z)$ admits an ...

**2**

votes

**0**answers

110 views

### Extension to real number system [closed]

Suppose you have equation involving a number $s$
$s^2+ 1 = 0$,
to solve it one needs to treat $s$ as complex number $s = \pm i$, and introduce $i$ as imaginary unit.
Now suppose you have equation ...

**5**

votes

**2**answers

491 views

### Holomorphy of a function with values in a Hilbert space

I asked the same question on MathStackExchange. EDIT: This question has now a open bounty worth +50 reputation on MSE.
Denote by $\mathbb C^\infty $ the Hilbert space $\ell^2 (\mathbb C)$. Fix $1\leq ...

**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

141 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 $|f|_r=\sup_{...

**3**

votes

**0**answers

115 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

131 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

104 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

281 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

46 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
$$\left(\...

**0**

votes

**1**answer

75 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} \right]\mathbf{x}_k\mathbf{x}_k^H\...

**1**

vote

**1**answer

89 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

333 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 $$\frac{\zeta'}{\zeta}\left(s\right)=\sum_{\left|t-\gamma\...

**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

508 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

114 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

70 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

133 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

164 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

237 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

448 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

94 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

160 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

106 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

90 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

114 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 \binom{z}{n}...

**8**

votes

**4**answers

385 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 ...