-1
votes
1answer
120 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 ...
5
votes
1answer
181 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
2answers
428 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
2answers
455 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
1answer
57 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
3answers
295 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
2answers
236 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
69 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 ?
3
votes
1answer
316 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 ...
6
votes
2answers
335 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
152 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
93 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
1answer
79 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
0answers
131 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 ...
2
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
171 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
147 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
179 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
1answer
61 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
2answers
361 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
288 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
0answers
190 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
0answers
180 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
1answer
306 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
0answers
144 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, $f(0)=0$, $f'(0)>0$. ...
0
votes
1answer
101 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
1answer
145 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 ...
11
votes
2answers
523 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
3answers
831 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
1answer
269 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
2answers
394 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
0answers
131 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
1answer
94 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
0answers
151 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( ...
6
votes
4answers
513 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 ...
11
votes
2answers
667 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, ...
12
votes
0answers
584 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
1answer
360 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
5answers
1k 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( ...
4
votes
0answers
247 views

Etymology of the O-notation for algebras of holomorphic functions

The notation $O(X)$ seems to be a quite standard notation for the algebra of all holomorphic functions on some connected domain in $\mathbb{C}^n$ (or a complex manifold). I would like to know where ...
3
votes
1answer
156 views

Can $-1/a_2$ belong to the range of a schlicht function $z+a_2z^2+\cdots$? Or is $-1/a_2$ necessarily an omitted value?

Is there an example of a schlicht function $f(z)=z+a_2z^2+a_3z^3+\cdots$, which is analytic and injective on the open unit disk $\mathbb{D}$, such that $-1/a_2$ belongs to the range $f(\mathbb{D})$? ...
15
votes
4answers
832 views

Why are lacunary series so badly behaved?

Hi! I just came across the Ostroski-Hadamard gap theorem, and while I can understand the proofs as well as the principle that the series $\sum_{n=0}^\infty z^{2^n}$ ought to have a singularity at ...
-1
votes
1answer
772 views

1895 Math Trip problem on primitive roots of unity

How to prove that if $\theta _1,\theta _2,\theta _3$ be the arguments of the primitive roots of unity, $\sum \cos p\theta = 0$ when $p$ is a positive integer less than $\dfrac {n} {abc\ldots k}$, ...
1
vote
3answers
496 views

Convergence of analytic covering maps to a covering map

Suppose a sequence of analytic maps $f_n: \mathbb{D} \to \mathbb{D}$ from the unit disk to itself, each of which is a topological covering map to its image, converges locally uniformly to an analytic ...
8
votes
1answer
378 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 ...
0
votes
2answers
752 views

Motivation behind defining the Ramification Divisor

I would like to understand what exactly is the motivation for defining the notion of a ramification divisor of a function. As I see the definition, If $f$ is a meromrophic function between two ...
1
vote
1answer
545 views

A question about Ahlfors's proof of modular function being a covering space of the twice punctured plane

I have a question about Ahlfors's proof of modular function being a covering space of the twice punctured plane .See Ahlfors' complex analysis, second edition, page 272. You can either explain or ...
6
votes
1answer
339 views

Analytic functions with algebraic Taylor coefficients at some point.

This question just came to my mind when reading the question When may Function (meromorphic) be expanded as power series with coefficients of integers Suppose $f$ is an analytic function on some ...
-1
votes
1answer
266 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
votes
2answers
324 views

Behavior of essential singularities in an 'open cone'

Picard's Big Theorem says that if a function $f(z)$ has an isolated essential singularity at a point $w$, then in every neighborhood of $w$, $f(z)$ hits every complex number infinitely many times, ...