# Tagged Questions

**-1**

votes

**0**answers

25 views

### Conformal map from a sector of unit disk onto upper half plane

How do we construct a conformal map from $\{z=x+iy,x>1/2,|x+iy|<1\}$ onto the upper half plane? My idea is first create a sector sending one of the two intersection points to infinity.Any help ...

**3**

votes

**2**answers

297 views

### Power series with funny behavior at the boundary

Consider a power series
$$
\sum_{n=0}^{\infty}a_nz^n
$$
where $a_n$ and $z$ are complex numbers. There is radius $R$ of convergence. Let us assume that is a positive real number. It is well known that ...

**8**

votes

**1**answer

200 views

### Different notions of convergence of complex subvarieties

Let $X$ be a smooth complex algebraic variety (or, better, complex analytic manifold). Let $\{C_i\}$ be a sequence of compact algebraic subvarieties (resp. analytic reduced subspaces) which converges ...

**1**

vote

**1**answer

137 views

### The locus of rational/elliptic curves on a special surface in $\mathbb{P}^3$

Let $P$ and $Q$ be two general polynomials of the same degree $d>5$. Consider the surface $S: z^2=P(x)Q(y)$ in $\mathbb{P}^3$ (after homogenization by the variable $w$). One can show that these ...

**0**

votes

**0**answers

67 views

### Boundedness and Convergence of a Complex sequence

Consider a dynamical systems over complex numbers $$ z_{n+1}=\frac{\alpha}{z_{n}}+ \frac{\beta}{z_{n-1}},\qquad n=0,1,\ldots $$
where the parameters $\alpha, ~\beta$ are complex numbers, and the ...

**6**

votes

**3**answers

149 views

### Is there a effective computational criterion to all periodic points of a rational function are repelling.

I came up with a question to know the fatou component of of some types of rational function. In some sense, I may need to give a computational criterion to existence of attracting periodic basin for a ...

**1**

vote

**1**answer

166 views

### System of quadratic complex equations

I want to solve this system of N non-linear equations without using a numerical method:
$x_{k}^{2}= \alpha_{k }+ \sum\limits_{m=1}^{N} (\beta_{km} x_{m} + \psi_{km} x_{m}^{*})$
With
$\left| ...

**1**

vote

**1**answer

298 views

### Infinite product's question

Given a pair of strictly increasing functions $f,g:\mathbb{N}\to \mathbb{N}$
define:
$P_N(f,g)\doteq \left(z\in \mathbb{C}\mapsto \prod_{i=1}^{f(N)}\left(1+\frac{z}{v_i(N)}\right)\in ...

**0**

votes

**1**answer

155 views

### Injective element of a commutative Banach algebra

A revision:
According to the comment of Nate Eldredge, in order to avoid the triviality, we revise the property $P$.
Assume that $A$ is a commutative unital Banach algebra. Its maximal ideal ...

**2**

votes

**2**answers

101 views

### Original article about a theorem of Cartan on iterations of analytic functions

I'd like to know in which paper of H. Cartan I could find the following theorem :
Let $\Omega$ be a connected, open and bounded subset of $\mathbb{C}$. Let $a \in \Omega$ and $f \in ...

**-2**

votes

**1**answer

132 views

### Holomorphic maps on $\mathbb{R}^{n}$ (for n not necessarily even)

Edit according to the comment of user36931 I remove the "motivation" from the previous version and I add an statement to the first question
We consider the following two classes of smooth maps on ...

**10**

votes

**2**answers

583 views

### Is there a holomorphic Morse-Bott lemma?

It asks for a generalization of the question in the post
Normal form for a holomorphic Morse function
Suppose $f$ is a holomorphic function on a complex manifold $M$ which has Bott type critical ...

**2**

votes

**2**answers

455 views

### $ 2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$ if and only if $f$ is linear

I know the following is a well-known result.
Let $D = B(0,1) \subset \mathbb{C} $ a disc, $f$ holomorphic on $D$. Show that $$ 2|f^{'}(0)| \le \sup_{z, w \in D} |f(z)-f(w)|$$
Furthermore, there is ...

**19**

votes

**2**answers

512 views

### Which smooth compactly supported functions are convolutions?

If $f,g$ are smooth functions with support in the interval $[-r,r]$ for some $r>0$, then their convolution $f*g$ is smooth with support in $[-2r,2r]$. My question is about the converse: Given ...

**0**

votes

**1**answer

66 views

### Cardinality of Picard exceptional values of holomorphic function without an isolated singularity

Picard's theorem says that if $f\colon D\to \mathbb{C}$ is an entire nonconstant function or holomorphic with an essential isolated singularity, then $\mathbb{C}\setminus f({D})$ is either empty or a ...

**7**

votes

**3**answers

337 views

### Summation of a series

I would like to sum the series
$$
\sum_{n=0}^\infty \frac{1}{(1+a^2 (n+1/2)^2) ^{3/2}} .
$$
It arose when trying to perform a calculation on superconductivity. In particular I am interested in its ...

**2**

votes

**2**answers

259 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

**0**answers

73 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 ?

**2**

votes

**1**answer

325 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

**2**answers

480 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

**0**answers

250 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

**1**answer

104 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 ...

**1**

vote

**1**answer

83 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 $\forall m\geq 1$.
...

**3**

votes

**0**answers

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 ...

**3**

votes

**3**answers

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

**2**answers

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

**1**answer

151 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

**1**answer

218 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
...

**1**

vote

**1**answer

74 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 ...

**9**

votes

**2**answers

377 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

**2**answers

298 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 ...

**0**

votes

**0**answers

225 views

### On uniform convergence of sequences of bounded holomorphic functions with formal convergence

At some point I needed to prove that some formal (iterative) construction yielded an actual convergent power series. To do so I was led to prove the following lemma, where $\mathcal{B}(\Delta)$ is the ...

**2**

votes

**0**answers

200 views

### Analytical continuation of electrostatic potentials

I'm having some trouble figuring out the properties with respect to analytical continuation of functions defined using an integral kernel. More particularly, I am working with the electrostatic ...

**0**

votes

**1**answer

335 views

### Question on Hartogs's Extension Theorem

Does Hartogs's extension theorem hold if one replaces the word holomorphic by analytic (of course still in several variables)?
For Hartogs's Extension Theorem see here:
...

**2**

votes

**0**answers

158 views

### Univalent functions with non-negative coefficients

Is anything non-trivial known about univalent functions with non-negative coefficients?
Let $U$ be the unit disc, and $f$ a univalent (=injective) holomorphic function, $f(0)=0$,
$f'(0)>0$. ...

**3**

votes

**0**answers

133 views

### Bounding an integral transform ouside a circle (or inside a strip)

Let $g$ be a symmetric unimodal probability distribution and $H$ be the right half plane.
We call
$$f(z) = \int_{-\infty}^\infty \frac{1}{z-i t}g(t)dt$$
the dispersion function of $g$.
Now, one can ...

**0**

votes

**1**answer

103 views

### Counting complex solutions on a disk.

I am looking for an example of an analytic map $f$ on the complex domain for which there exist $r>0$ such that for $\gamma=\{z:|z|=r\}$, a.s. for $\theta \in [0,2\pi)$ we have that
...

**1**

vote

**1**answer

149 views

### Can a locally defined holomophic function which vanishes on a subvariety $V$ be written in terms of globally defined polynomials vanishing on $V$?

Given a complex algebraic, affine variety $X$ and a subvariety $V$, let $I_V$ indicate the ideal of polynomials on $X$ with vanish on $V$.
Given an open subset $U$ of $X$, is it true that the ideal ...

**12**

votes

**2**answers

545 views

### Belyi functions on non-compact surfaces; or: Building Riemann surfaces from equilateral triangles

Some background on (compact) Belyi surfaces
$\newcommand{\Ch}{\hat{\mathbb{C}}}$
A compact Riemann surface $X$ is called a Belyi surface if there exists a branched covering map $f:X\to \Ch$ such that ...

**12**

votes

**3**answers

883 views

### If a formal power series over the complex numbers satisfies a polynomial identity, does it imply that the power series has a radius of convergence?

Let $ P(z) $ be a $\textit{formal}$ power series in $z$ that a priori may not have a non zero radius of convergence. Assume that $P(0) =0$.
Let $\Phi(w,z)$ be a polynomial in two variables, that ...

**2**

votes

**1**answer

340 views

### Constructing the imaginary part of a holomorphic function

Hallo,
Let $f: U \rightarrow \mathbb{R}$ be a analytic function, where $U \subset \mathbb{C}^{n}$ is a open set (paracompact, starshaped or convex i.e. sufficiently nice). Does there exist a function ...

**11**

votes

**2**answers

418 views

### A “holomorphic” Peano curve?

A Peano curve is a continuous map $[0,1]\to [0,1]^2$ whose image is the whole square.
I would like to know if on can obtain "holomorphic" Peano curves. Namely, is it possible to find a continuous ...

**5**

votes

**0**answers

135 views

### Perturbations of zero-dimensional algebraic varieties

Let $P(z,w):\mathbb C^2\to\mathbb C$ be a certain polynomial, and consider $p(s,t)=P(e^{is},e^{it}):\mathbb T^2\to \mathbb C$ its restriction to the real torus. Assume generic situation, so by ...

**0**

votes

**1**answer

95 views

### Bloch type function

I would like to know whether there exists an analytic function $f$ on the unit disk such that $$\sup_{|z|<1}|f(z)|(1-|z|^2)<\infty$$ and for every $|a|=1$, $$\limsup_{z\to ...

**0**

votes

**0**answers

156 views

### Harmonic Function?

Hi,
Let $\varphi : U \rightarrow \mathbb{C}$ be a holomorphic function, where $U \subset \mathbb{C}^{n}$ containing $0$. Is the function $u(x_{1}, ..., x_{n}, y_{1}, ..., y_{n}) := Im( ...

**7**

votes

**4**answers

574 views

### Seeking a Geometric Proof of a Generalized Alternating Series' Convergence

Let $z \in \mathbb{C} \backslash \lbrace 1 \rbrace$ with $|z| = 1$. We consider the following infinite series, which necessarily converges:
$$S(z) := \sum_{n = 1}^{\infty}\frac{z^n}{n}$$
Note that ...

**12**

votes

**2**answers

759 views

### Determining rational functions by their critical points

Fix an integer $d > 1$ and $2d-2$ points $P_1, \ldots, P_{2d-2}$ in the Riemann sphere (not necessarily distinct). Thanks to the work of Eisenbud and Harris on limit linear series (Inventiones, ...

**13**

votes

**0**answers

625 views

### What is the appropriate setting for Cauchy's Integral Formula?

For a $C^1$ function $f:U\to\mathbb{C}$, where $D\subset U\subseteq \mathbb{C}$ with piecewise $C^1$ boundary $\partial D$, we have the following generalized Cauchy integral formula:
$$
f(\zeta) = ...

**1**

vote

**1**answer

394 views

### How to study the nonregular part of a finite branched holomorphic covering?

A finite branched holomorphic covering is a holomorphic map $f : V \to W$
between holomorphic varieties $V$ and $W$ such that
$f$ is a finite branched covering (in the topological sense)
There is a ...

**2**

votes

**5**answers

2k views

### Why $\partial$ and $\bar{\partial}$ defined in that way (the Wirtinger derivatives)?

For $\mathbb{C}$-valued functions, why are $\frac{\partial}{\partial z}$ and $\frac{\partial}{\partial \bar{z}}$ defined as
$$
\frac{\partial}{\partial z}=
\frac{1}{2}\left(
...