Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange
Join us in building a kind, collaborative learning community via our updated Code of Conduct.

Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.

-2
votes
0answers
54 views

Property of a polynomial whose zeros lie in the half plane $\Re{(z)}\geq 1$

If $P(z)=\prod_{k=1}^n(z-z_k)$ with $\Re(z_k)\geq 1, 1\leq k\leq n$ then prove or disprove that $$\max_{|z|=1}|P'(z)|\leq \sum_{k=1}^n\frac{1}{1+\Re(z_k)}\max_{|z|=1}|P(z)|.$$ This is an open problem....
0
votes
0answers
90 views

Does Hartogs's Theorem for complex-analytic functions hold for real-analytic functions? [duplicate]

Recall a very famous theorem due to Hartogs for complex analytic functions of several variables. Hartogs's Theorem Let $f$ be a holomorphic function on a set $G \setminus K$, where $G$ is an open ...
3
votes
1answer
188 views

Chow theorem in $\mathbb{C}^2$

I have the following question the answer to which I cannot find in the literature (but it must have been studied): Suppose that $M\subset\mathbb{C}^2$ is a real surface which may locally be written ...
-3
votes
0answers
53 views

On Fourier transforms of even functions [on hold]

Suppose that $F$ is a real-valued Fourier transform of the function $G$, and that $F$ is even. That is $F(x)=F(-x)$ for all $x \in \mathbb{R}$. Does this necessarily mean that $G$ is also even/odd ? ...
1
vote
0answers
68 views

A circle separating the critical points

A circle $C$ is said to be a separating circle for a set $S=\{z_1,z_2,\cdots,z_n\}$ if either there are points from $S$ in the interior and the exterior of the circle $C$ or all points of $S$ are on $...
4
votes
0answers
112 views
+100

Sobolev space under Mellin transform

The Mellin transform is known to be an isomorphism see wikipedia between $M:L^2(0, \infty) \rightarrow L^2(-\infty, \infty)$ where $$M(f):= \frac{1}{\sqrt{2\pi}}\int_0^{\infty} x^{-\frac{1}{2} + is} ...
0
votes
0answers
70 views

A property of a polynomial of degree 2

If $P(z)=(z-z_1)(z-z_2)$ where $z_1,z_2$ are any complex numbers with $|z_1|\geq 1, |z_2|\geq 1.$ Then let us try the proof for a naive inequality $$\max_{|z|=1}|P'(z)|\leq \max_{|z|=1}|P(z)|.$$ I am ...
3
votes
0answers
74 views

An inequality with rotation

Let $P(z)=\sum_{m=0}^na_mz^m$ be a polynomial of degree $n$ having no zeros in $|z|<1,$ then for any complex number $\alpha$ with $|\alpha|=1,$ is it true on $|z|=1$ that $$\left|\alpha zP'(z)+zP'...
2
votes
0answers
72 views

Universal cover of ladderly puntured complex plane

The twice punctured complex plane $\mathbb{C}-\{0,1\}$ has as its universal cover the upper half plane via elliptic modular function. I am looking for the constructions of the covering map from the ...
1
vote
1answer
77 views

Zeros of Multivariate Complex Functions [need reference]

I am looking for a good accessible reference that would summarize properties of zeros of complex analytic functions. For my purpose, it would be interesting to see a discussion on the following ...
0
votes
0answers
41 views

Finding positive powers of a Laurent series

Consider the functions $G(z)$ defined as a Laurent series $$ G(z)=\sum_{n=-\infty}^\infty g_n z^n, \; z > 0, $$ and $G(1)=0$. Also consider the function $f(z)$ defined through $G(z)$ as $$ f(z) =...
0
votes
0answers
33 views

Holomorphic function has radial limit at z=0 [duplicate]

Let f(z) be a holomorphic function defined on C-{0} satisfying f(r*e^iθ)→0 as r→0 for each θ∈R. Must z=0 be a removable singularity?
2
votes
0answers
35 views

Is singular Cauchy operator bounded in Morrey spaces?

The singular Cauchy operator is defined by $$S_\Gamma :f \to \int_\Gamma \frac{f(\xi)}{\xi-z} d\xi , z\in \Gamma.$$ Is this operator bounded in Morrey spaces and weighted Morrey spaces? i.e. is there ...
2
votes
0answers
89 views

On a real part of a series with complex numbers

Let $P(z)=\sum_{m=0}^na_mz^m $ be a polynomial of degree $n$ having all its zeros in $|z|\leq 1.$ Then what is the best value for 'L' in $$\Re\left(\sum_{k=1}^nP(zw_k)\frac{w_k}{(w_k-1)^2}\right)\geq ...
2
votes
4answers
189 views

Complex differential equations

I'm looking for a gentle an concise introduction to complex-variable differential equations. Eventually, I need to look at complex PDEs, but I assume one starts with complex ODEs. Mostly, I'm just ...
1
vote
0answers
54 views

Derivative of complex matrix pseudo inverse with respect to real and imaginary components

I have a complex non-square matrix $\mathbf{Y}\in\mathbb{C}^{n \times m}$ whose inverse I compute using the Moore-Penrose pseudo inverse, $\mathbf{Z}=\mathbf{Y^+}$. I am interested in evaluating the ...
1
vote
0answers
50 views

Non-compact analogue of Hartog's extension theorem?

Suppose a function $f(z,w)$ is analytic in the open polydisk $\Delta^2$ with $\Delta = \{z \in \mathbb C | |z| < 1 \}$. I am interested in the limit $f(z,w)$ as $w \to 1$. This limit may be ...
1
vote
0answers
204 views

Induction principle on proving an inequality

If $P(z)$ having no zeros in $|z|<1,$ then $$\frac{\max_{|z|=1}|P'(z)|}{\max_{|z|=1}|P(z)|}\leq \frac{n}{2}.$$ Can we prove this by induction on $n$? or is there any alternative way? Attempt at ...
1
vote
1answer
109 views

The generalization of Hartogs' Theorem

I know a version of Hartogs' Theorem in the book An introduction to complex analysis(p 30) by Hormander, namely Hartogs' Theorem when K compact with complement being simply connected I also have ...
3
votes
1answer
87 views

compare N(f,a,r) with T(f,r)

I'm reading William Cherry and Zhuan Ye's book 'Nevanlinna's theory of value distribution, the second main theorem and its error terms'. In Section 1.12, they explains why $N$ and $T$ is used in ...
5
votes
2answers
208 views

Vector valued disc “algebra”

I am interested in a vector-valued form of the disc "algebra" (which in this setting is not in general an algebra, hence the scare quotes). Let $E$ be a Banach space, and let $A(\mathbb D,E)$ be the ...
2
votes
0answers
83 views

A generalized Cauchy type functional equation

Let $(S,+)$ be an abelian semigroup . Let $f:S \to \mathbb C$ be a function such that for some positive integer $n>1$, $f(x+y)^n=(f(x)+f(y))^n,\forall x,y \in S$. Then is it true that $f(x+y)=f(x)...
0
votes
1answer
150 views

An inequality that involves integrals

Assume that $g(re^{it}),$ and $h(re^{it})$ are smooth positive functions defined on the annulus $A=A(R,1)=\{z: R<|z|<1\}$. Assume also that $\int_0^{2\pi}h(re^{it})dt\ge 2\pi c$ for every $r\in(...
2
votes
1answer
157 views

fast algorithms for external angle computations

Two related problems related to the complex quadratic polynomial $f_c(z) = z^2 + c$ and Mandlebrot and/or Julia sets: find an external angle $\theta_c$ for a complex point $c$ find a complex point $...
2
votes
1answer
166 views

Roots of unity and an extremal problem [closed]

I want to determine the subset of $m$ members ($m < n/2$) of the set $e^{i 2\pi k/n}, \ \ k=0,\dots, n-1$, so that the absolute value of its sum is maximal.
13
votes
4answers
2k views

Teaching Prime Number Theorem in a Complex Analysis Class for Physicists

This is a question about pedagogy. I want to sketch the proof of the prime number theorem or any other application of complex analysis to number theory in a single lecture, in a complex analysis ...
3
votes
0answers
72 views

Reference request: basics about modular curves

Where can I find a reference (with carefully written proofs) for basic facts about modular curves? Namely: Congruence subgroups The open modular curve $Y_\Gamma$ admits the structure of a Riemann ...
2
votes
2answers
207 views

Does Bergman metric induce the standard topology?

I am a physics student and am interested in the study of invariant metrics. I have searched several textbooks, including those fat books of Krantz, but the following concern seems not to be mentioned ...
-3
votes
1answer
116 views

An interesting phenomenon of the analytic continuation of Riemann zeta function [closed]

It is known that $$\Gamma (s) \zeta (s)=\int_0^{\infty} \frac{x^{s-1}}{e^x-1}dx$$ this function is valid only for $\Re{s}>1$. However, if we ignore this restriction, and integrate by using $$\frac{...
1
vote
1answer
57 views

Conditions to obtain a real logarithm of a unitary unimodular complex matrix?

The problem statement is the following: $$U=\exp\{iV\}$$ where $U$ is a unitary unimodular matrix of the following form: $$U=\begin{bmatrix}u_1+iu_2&u_3+iu_4\\-u_3+iu_4&u_1-iu_2\end{bmatrix}...
3
votes
1answer
139 views

On the values of an entire function

Let $0<q<1$ and consider the entire function $f(z)=\displaystyle \sum_{k=0}^\infty q^{k^2}z^k$. For $a>1,$ denote $m_j=f(a^j),\; j=0,1,2,\dots.$ Question: Does there exist an entire function ...
4
votes
0answers
131 views

Bezout theorem for germs of holomorphic functions

UPDATE. It was pointed out by @Dmitri that two smooth curves given by $f=y$ and $g=y+x^k$ in $\mathbb C^2$ provide a simple counterexample. Let $f_1, \ldots, f_p, g_1, \ldots, g_q$ be germs of ...
0
votes
0answers
116 views

An open inequality

If $P(z)=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+z^n$ is a polynomial of degree $n\geq 1$ having all its zeros in $|z|\leq 1,$ then for all $z$ on $|z|=1$ for which $P(z)\neq 0$ is it true that $$\text{Re}\...
5
votes
1answer
118 views

Holomorphic functions with equal inverse images of unit circle

Let $f,g:\mathbb{C} \to \mathbb{C}$ be holomorphic and have the property $f^{-1}(S)=g^{-1}(S)$ where S is the unit circle centered at 0. What can be said about $f$ and $g$.
0
votes
1answer
172 views

Exponential Sequence of Sheaves

Let $(X, \mathcal{O}_X)$ be a complex analytic space in the sense of Grauert, i.e., a $\mathbb{C}$-analytic ringed space which is locally isomorphic to a local model. We may assume that $X$ is a ...
1
vote
1answer
77 views

Gluing locally defined continous functions over complex domain

This is a cross-post to the question I asked at MSE over almost a month ago. Suppose $n, l, m \in \mathbb N$ and $n \ge l > m$. Let $T: \mathbb C \to \mathcal M(n \times l; \mathbb C)$ be ...
1
vote
1answer
84 views

positive real matrix-valued function as linear combination of positive-real functions

In my question I am considering $z$ as the complex variable in the $z$-transform $X(z)$ of a discrete-time sequence $x[n]$: I have $M$ square complex matrices $\mathbf{R}_m$ of size $N\times N$. I ...
1
vote
1answer
122 views

zeros of sums of complex exponential functions

Let $a_i,b_i$ be $n(\geq 2)$ non-zero real numbers. Assuming that $\sum_{i=1}^n a_ie^{\sqrt{-1}b_i x}=1$ has infinite real solutions for $x$, prove or disprove that $b_i(1\leq i\leq n)$ is linearly ...
3
votes
0answers
138 views

best-possible inequalities for hypergeometric functions

In what follows, let $n$ be a positive integer and $0<a<1/2$. I am interested in the Gauss hypergeometric functions, $_{2}F_{1}( -n, -n-a; 1-a; z)$. Notice that these are polynomials, if that is ...
1
vote
0answers
125 views

The comparison of certain modules arising from the Cauchy-Riemann differential operator

Let $\Gamma=C^{\infty}(\mathbb{R}^2)$ be the space of all smooth complex valued functions on the plane. We define the following Cauchy Riemann differential operator $D$ on $\Gamma$: $$D:\Gamma \...
0
votes
4answers
309 views

On the real part of the Riemann zeta function inside the critical strip

Denote by $\zeta$ the Riemann zeta function. Does $\Re\zeta(s)$ ever vanish for $\frac{1}{2}<\Re(s)\leq 1$ ?
4
votes
2answers
293 views

Is Riemann zeta function injective in some strips $a<\Re(s)<b$, where $0\leq a<b \leq 1$?

Or more generally, are L-functions injective in some strips $a<\Re(s)<b$, where $0\leq a<b \leq 1$?
1
vote
1answer
133 views

Generating series of rational$\times \exp($rational$)$

It is known that rational functions $f\in \mathbb C(x)$, $0$ not a pole, are the sum of generating series $\sum_{n\geq 0} a_nx^n$ where $(a_n)_n$ is solution of a linear recurrence with constant ...
12
votes
2answers
738 views

Algebraic vs analytic normality

Let $X$ be a complex algebraic variety. We can ask if $X$ is normal as an algebraic variety, but also, if its analytification is normal as a complex analytic space. Is there a relationship between the ...
4
votes
0answers
80 views

Continuous function on a complex space that is holomorphic on the complement of a closed subspace

Let $X$ be a complex analytic space and $Y\subseteq X$ a closed complex subspace. Suppose that $f:X\to\mathbb{C}$ is a continuous function that is holomorphic on $X\setminus Y$. Is $f$ holomorphic on $...
1
vote
0answers
43 views

Bound for truncation error of continued fraction for $E_1(z)$

Let $z \in \mathbb C \setminus(-\infty,0)$. It is known that $$E_1(z) = \cfrac{e^{-z}}{z+\cfrac{1}{1+\cfrac{1}{z+\cfrac{2}{1+\cfrac{2}{z+\cfrac{3}{1+\cdots}}}}}}.$$ For example, see http://functions....
10
votes
0answers
383 views

Witt's proof of Gelfand-Mazur / Ostrowski's Theorem

Previously asked on Math Stackexchange without answers. Background: As sort of a hobby, Ernst Witt gave extremely short proofs for famous theorems. This question is about his six-line proof of the ...
3
votes
1answer
156 views

About some positive elements in a group von Neumann algebra

Let $G$ be a (discrete) torsion free group with identity $e$. Recall that for an element $\alpha=\sum a_gg$ in $\mathbb C[G]$ (complex functions on $G$ with compact support), $\alpha^*$ is defined to ...
4
votes
1answer
143 views

$L^1$ norm of Littlewood polynomials on the unit circle

A Littlewood polynomial is a polynomial with coefficients from $\{ 1, -1\}$ and the set of Littlewood polynomials with degree $n$ is denoted by $\cal{L}_n$. I'm interested in a "good" lower bound on ...
6
votes
1answer
226 views

An Optimization Problem with Complex Variables, regarding Eigenvalues of Circulant Matrices

Let $S$ be a finite subset of the complex unit circle and $1 \in S$. For each $n \in \mathbb N $, define $f_n\colon S^{n-1}\to\mathbb R$ by $$f_n(x) := \sum_{w^{n}=1}|x_1w+ x_2w^2\cdots+x_{n-1}w^{n-1}...