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0
votes
2answers
85 views

Subharmonic function on a twice punctured complex plane

is the twice punctured complex plane parabolic or hyperbolic? In this sense: does $\mathbb{C}-\{0,1\}$ admit a nonconstant, negative, subharmonic function? Thanks,
0
votes
1answer
135 views

Relation between the eigenvalue density and the resolvent?

Disclaimer: This is a cross-post from Math Underflow. Given that there is little activity on the subject (random-matrice) on the aformentioned site, and given that many interesting discussion on this ...
1
vote
0answers
68 views

How to find number of points at infinity of a Riemann surface

Let $X \subset \mathbb C^2$ be a Riemann surface with boundary $\partial X \subset \mathbb C^2$ and without compact components. Let $\bar X = X \cup \{p_1,\ldots,p_N\} \subseteq \mathbb CP^2$ be its ...
5
votes
1answer
231 views

Asymptotic value of a multivariate integral

The following question is a simple case of a type of problem that occurs in combinatorial enumeration problems. Define $$F(x_1,\ldots,x_n) = \frac{1}{(2\pi)^{n/2}}\exp\biggl( -\frac12\sum_{j=1}^n ...
8
votes
1answer
836 views

If $f$ is $C^{\infty}$ and $f^2$ is analytic, then $f$ is analytic

Assume that $f:\mathbb{C}^n\rightarrow \mathbb{C}$ is a $C^{\infty}$ function such that $f^2$ is (complex) analytic. Then one can show that $f$ is analytic. (Note: Liviu Nicolaescu and Alexandre ...
2
votes
1answer
88 views

Convolution-type operator for series

Suppose $f(z)=\lambda(a_1z+a_2z^2+\cdots)$ is holomorphic in $\{|z|<1\}$ with $\lambda>0$. For each $d\geq 1$ , I am trying to define an operation, $\star_d$ , so that $f(z)\star_d ...
0
votes
1answer
114 views

Residues and Mittag-Leffler sequence

Let $X$ be a compact Riemann surface, $\omega$ a meromorphic differential on $X$ and $f$ a meromorphic function on $X$ with poles only over the points $P_1,\dots,P_d$. The product $\;f\cdot\omega\;$ ...
2
votes
2answers
256 views

Surjective entire functions without critical points

It is easy to construct surjective locally univalent holomorphic functions $f: {\mathbb D}\to {\mathbb C}$, where ${\mathbb D}$ is the open unit disk. I am pretty sure that the answer to the ...
1
vote
0answers
124 views

Extensibility of real analytic function of several variables to complex domain

My question relates to the extensibility of a real analytic function of several variables to a specific complex domain. In order to formulate the question, let me define the following complex ...
1
vote
2answers
171 views

Analytic continuation for PI(1+z^(4^n))

How to do analytic continuation for following function? $$f(z) = \prod_{n=0}^{+\infty} {(1+z^{4^n})}$$ Evidently it satisfies $f(z)f(z^2)=\dfrac{1}{1-z}$...
1
vote
0answers
72 views

analytical behaviour of holomorphic functions on boundary

Is there any survey about criterions for a holomorphic function to be analytical on the boundary of a pseudo convex domain on which it is defined ?
0
votes
1answer
115 views

Existence of a holomorphic function with specific caracteristics

Is it possible to find a holomorphic function $f : D \rightarrow \mathbb{C}$ where $D$ is the $\mathbb{C}$ open unit disk such that: $f$ is continuous in $\overline{D}$ $f (\partial D)\subset ...
2
votes
1answer
323 views

Finding the residue for a complex function defined using an infinite product

Suppose we define the infinite product $\displaystyle \prod_{n=1}^{\infty} (1+a^{-ns})^{a_n}$, where $a_n$ is some given sequence of positive integers. Is there a way, supposing there is a pole at ...
5
votes
2answers
402 views

An extension of Morera's Theorem

Morera's Theorem states that If $f$ is continuous in a region $D$ and satisfies $\oint_{\gamma} f = 0$ for any closed curve $\gamma$ in $D$, then $f$ is analytic in $D$. I have two questions: ...
1
vote
0answers
243 views

complex contour integral calculation after Möbius transformation

Good day to everyone. In my scientific research I've got stuck with a contour integration problem. I would like to evaluate the following integral: $$I=\int_0^{\infty } \frac{e^{\frac{\alpha -\mathrm ...
1
vote
1answer
102 views

Monotonicity and perturbation of $J$-holomorphic curves

In a symplectic manifold $(M, \omega)$, $J$-holomorphic curves are special minimal surfaces if $J$ is compatible with the symplectic structure and the metric is induced by $\omega$ and $J$. Minimal ...
-2
votes
1answer
157 views

Howto plot a specific complex function [closed]

We need to plot the real and imaginary parts of a complex function $k(\omega)$, and cannot find a good way to do this without using "ad hoc tricks." Definitions $k$ is a complex-valued function ...
1
vote
1answer
81 views

Normal family and arithmetic progression

It is basic fact that for a holomorphic (or mermorphic) map $f$, the family of iterates $\{f^n\}_{n=1}^{\infty}$, is normal if and only if $\{f^{mn}\}_{n=1}^{\infty}$, is normal $\forall m\geq 1$. ...
2
votes
0answers
68 views

Introducing new poles

Suppose we have a contour integral of an entire function along a line in the plane (or perhaps a line segment). I am looking for examples of such integrals that are computed by first altering the ...
3
votes
0answers
138 views

elementary proof for existence of point with minimal period 2 for entire function

Fatou proved a very interesting result: for a transcendental entire function $f$, the second itarate $f^{2}$ has at least has one fixed point. (Using the technique of Picard theorem) This result ...
7
votes
1answer
314 views

Abscissa of convergence of Dirichlet series

Let $D(s)$ be a Dirichlet series with abscissa of convergence $\sigma_c=\sigma_a$. Does it follow that the Dirichlet series defined by $P(s)=D(s)\bar D(\bar s)$ has the same abscissa of convergence? ...
1
vote
1answer
198 views

Some questions about inner functions

To avoid trival cases, we assume that $f$ is neither a constant nor a finite Blaschke product. Two celebrated theorems of Frostman say that $f_a(z)$ is actually a Blaschke product for every ...
0
votes
0answers
90 views

Derivative of a function related to Dedekind zeta function

Lef $K$ be an algebraic number field of degree $[K:\mathbb{Q}]=n$. For simplicity suppose $K$ is totally real. Define $f(s) = \zeta_K(s) \zeta(1-s)^{n-1}$ where $\zeta = \zeta_{\mathbb{Q}}$. From the ...
0
votes
1answer
163 views

How to characterize the value which assumed by an inner function infinitely often but is an asymptotic value?

The Frostman Shift of an inner function at the value which assumed infinitely and is not an asymptomatic value is an infinite Blaschke Product. But how to charaterize it when it is an asymptotic ...
0
votes
1answer
72 views

The influence of eigendecomposition on the periodicity of a (rank 2) Hermitian matrix (of functions)

Let $\boldsymbol{R}(u,v);~u,v\in\mathbb{R}$ be a Hermitian matrix (of Hermitian functions) with entries \begin{equation} r_{ij}(u,v) = 1 + Ae^{-2\pi i \phi_{ij}(ul_0 + vm_0)}; ...
1
vote
1answer
97 views

Has a universality theorem been proved for the Davenport-Heilbronn L function?

The question is in the title: has a universality theorem in the sense of Voronin been proved for the Davenport-Heilbronn function, or do we expect such a theorem to hold true only for L functions that ...
2
votes
1answer
147 views

Inequality for certain analytic functions

Suppose that $f(z)$ is analytic for $\Re(z)>0$ and it is positive on the real axis. Suppose that for some $y_0$ we have $$ \left| {f\left( {x + iy} \right)} \right| \le f\left( x \right) $$ for any ...
3
votes
3answers
1k views

Does the inverse Laplace transform of the square root exist?

Does the inverse Laplace transform, defined by the integral, \begin{equation} F(t) = \mathscr L_s^{-1}\left[\sqrt s\right](t) = \int_{c - i\infty}^{c + i\infty} \sqrt s ~e^{-st} ds \end{equation} ...
0
votes
2answers
177 views

A Functional Equation concerning analytic functions

Let $P$ be a polynomial and suppose $f : \Bbb{C}\longrightarrow \Bbb{C}$ is a non-constant analytic function such for all $z \in \Bbb{C}, f(z) = f(P(z))$. Clearly when $P$ is linear we can find such ...
3
votes
1answer
150 views

The Integral Trick and An Equality in Nakajima's Lecture

In Nekrasov et al's series papers MNS, they calculate such kinds of integral $$\frac{E_1 E_2}{N(2\pi i)^N(E_1+E_2) }\oint d\phi_1 \wedge d\phi_2\wedge ...\wedge d\phi_N \prod_{i<N} (-\phi_i) ...
1
vote
1answer
206 views

Two different forms of Schwarz-Christoffel-Mapping of unit disk to rectangle. Are they identical?

I found two different equations for the Schwarz-Christoffel-mapping of a unit disk to a rectangle (which are the general form of the SC-mapping, I guess). The first, e.g. from Link, page 20, is ...
0
votes
2answers
172 views

Convergence of Dirichlet series (“at the boundary”)

I apologize if this is something standard and/or elementary, but I was unable to find anything relevant via Google. Consider a Dirichlet series $$ f(s) = \sum_{n=1}^\infty \frac{a_n}{n^s} $$ and ...
1
vote
1answer
268 views

Questions about expansion of $f(x)=\sum_{i=1}^{\infty} a_i x^i$

In complex field, assume $f(x)=\sum_{i=1}^{\infty} a_i x^i$ where $a_i \in {\bf N}$ or $a_i = 0$, and $f(x)$ converges in an area. Question 1: are there $$f(x)=p(x)+\sum_{i=1}^{\infty}r_i(x), $$ or ...
0
votes
2answers
246 views

Poisson inequality for subharmonic functions

This is probably a very basic matter, but I am looking for a proof of the Poisson inequality for subharmonic functions, which reads $$\varphi(r \mathrm{e}^{\mathrm{i} \theta})\leq\frac{1}{2\pi} ...
1
vote
1answer
73 views

Finite construction of lacunary functions using algebraic and certain analytic operations

Algebraic functions have a discrete set of singularities. Lacunary functions, e.g. $f(z)=\sum_{n=0}^\infty z^{2^n}$, have a continuum of singularities at every point of the boundary of their disk of ...
6
votes
1answer
266 views

interpolation with derivative of rational fraction

Studying a problem in conformal geometry, I am facing to the following interpolation problem. Let $P$ and $Q$ two coprime polynomials. Then let $A$ and $B$ two coprime polynomials such that ...
3
votes
0answers
108 views

What is the relationship between complex time singularities and UV fixed points?

In this paper it is described how the turbulent kinetic energy spectrum and the flatness (a measure for intermittency) are governed by the position of the (dominant) singularities of the solutions of ...
2
votes
0answers
199 views

computing a certain contour integral [closed]

I want to compute an integral along a vertical line segment. The function I'm integrating involves the zeta-function, and usually the way such integrals are done treats the line segment as one side ...
2
votes
1answer
149 views

Approximation Runge's Theorem

Let $X$ be a Riemann Surface and $K$ a compact subset of $X$. Every holomorphic function in $K$ be uniformly approximable on $K$ by holomorphic functions on $X$ if $X-K$ have no connected component ...
0
votes
1answer
415 views

construct a power series with infinitely many zeros in the complex plane, bounded coefficients???

Hi all. I want to construct a power series $F(z)=\sum_{n=0}^\infty c_nz^n$ centered at zero and with finite radius of convergence in the complex plane, and which has infinitely many zeros (in its ...
3
votes
0answers
209 views

The functional equation of Hofstadter's Q sequence

Hofstadter's Q sequence is defined by $Q(1) = Q(2) = 1$ and $Q(n) = Q(n-Q(n-1)) + Q(n-Q(n-2))$ for $n \geq 3$. So far hardly anything on this sequence has been proved -- not even that $Q(n)$ is ...
9
votes
2answers
373 views

Constructing Riemann maps using Brownian motion?

There's a relation between two-dimensional Brownian motion and conformal maps, see e.g. Thurston's answer to this question. Given two non-empty simply-connected domains $U$ and $V$ in the complex ...
2
votes
2answers
243 views

on completeness of R_mn, the set of all rational functions of type (m,n)

It is known from finite dimensionality of $P_r$, the space of all polynomials of degree less than or equal to $r$, that $P_r$ is complete with respect to uniform norm. Considering ...
2
votes
2answers
297 views

What is known about this product?

I bet the product $$ \prod_{n=2}^\infty\frac 1 {1-n^{-s}}, $$ which is convergent for ${\rm Re}(s)>1$, has been studied before. Can it be analytically extended across the line ${\rm Re}(s)=1$? If ...
5
votes
1answer
210 views

Is the homeomorphism class of a connected open set of C determined by its fundamental group?

Let $U,U'\subseteq\mathbf{C}$ be two connected open sets such that $\pi_1(U)\simeq\pi_1(U')$. Q: Does this imply that $U$ is homeomorphic to $U'$? In the case where the $\pi_1$'s are trivial then ...
6
votes
1answer
129 views

Is the dual of $A^1(\Omega)$ known for arbitrary domains ?

Let $\Omega$ be a domain in the complex plane, and $A^1(\Omega)$ be the space of integrable holomorphic functions on $\Omega$ equipped with the $L^1$ norm (it is called the Bergman space). If $\Delta ...
1
vote
0answers
80 views

Pointwise bounds on Hardy space functions with regular boundary behaviour

Let $H^2$ denote the Hardy space on the strip $S:=\{z\in{\mathbb C}\,:\,0<\Im z <1\}$ (or the upper half plane), i.e. $H^2$ consists of all holomorphic functions $f:S\to\mathbb C$ such that for ...
2
votes
1answer
350 views

Do there exist transcendental numbers which are not hypertranscendental?

A complex number is said to be hypertranscendental if the one is not a zero of any entire function with all rational Maclaurin coefficients. Does there exist a transcendental number which is not ...
1
vote
2answers
261 views

Any closed form for series like $F(x)=\Sigma_{i=p}^{\infty}x^p$,p is prime$?

Any closed form for series like $$F(x)=\Sigma_{i=2}^{\infty}x^p,\text{p is prime}$$ or $$F(x)=\Sigma_{i=0}^{\infty}x^{i!}$$? More generally,we can obtain a power series from decimal expansion of a ...
11
votes
2answers
699 views

Anything special (historical?) about surface $x\cdot y\cdot z\ +\ x+y+z=0$?

QUESTION I wanted to introduce and develop the complex logarithm from scratch. As the result I've arrived a couple of months ago at the following identity after which the road to complex logarithm is ...