# Questions tagged [cv.complex-variables]

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

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### Residues of Zeta-like Function

I'm looking for the residues of the following function $$s \mapsto\sum^\infty_{m,n =1} (m+n) \left[ amn + (m-n)^2 \right]^{-s}$$
at $s=\frac{1}{2}$ and $s=\frac{3}{2}$, where $a$ is some real positive ...

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**1**answer

366 views

### Estimating the derivative of a polynomial on the unit circle

Let $P(z)=\sum_{k=0}^na_kz^k $ be a polynomial of degree $n$ and $z_k (1\leq k\leq n)$'s be $n$th roots of $-1$. Then when $\theta=0$ the inequality
$$|P'(e^{i\theta})|\leq \frac{4}{n}\left|\sum_{k=...

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144 views

### Extremal polynomial majorants of $\log{|f|}$: a multivariate extension of a theorem of Carneiro and Vaaler

Carneiro and Vaaler have proved, as an application of their work on Beurling-Selberg extremal majorants, that for any non-zero complex polynomial $f(z) \in \mathbb{C}[z]$, the infimum value of the ...

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**2**answers

189 views

### Unramified map of Riemann surfaces

Let $f:S \to T$ be a surjective, unramified, holomorphic map between connected Riemann surfaces. If $S$ is not compact is it always true that $f$ is a covering?
This is of course true if $S$ is ...

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175 views

### Analytic Continuation of Zeta-like function

Reading a paper about eta invariants I came across a zeta-like function.
I'm looking for the analytic continuation of $$\sum_{k=1}^\infty k(k+a)^{-s}$$ at $s=0$, where $a$ is positive.
In the paper ...

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**1**answer

119 views

### Estimate for radius of convergence of solutions given by Cauchy-Kovalevskaya Theorem

I'm sure you can extract it from the proof, but does anyone know of a reference where the radius of convergence (in terms of radius of convergence of the initial data and PDE) of the solution given by ...

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105 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 ...

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267 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 ...

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82 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 $...

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168 views

### 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} ...

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147 views

### An inequality with rotation

Let $P(z)=\sum_{m=0}^na_mz^m$ be a polynomial of degree $n\geq 1$ 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'(...

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**1**answer

113 views

### Relation between infinite product and regularized product

For a positive sequence $0\le\lambda_{1}\le\lambda_{2}\le\cdots$, consider an infinite product
\begin{equation*}
\prod_{i=1}^{\infty}\lambda_{i}:=\lim_{n\rightarrow\infty}\prod_{i=1}^{n}\lambda_{i}...

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89 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 ...

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97 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 ...

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39 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 ...

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100 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 ...

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219 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 ...

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82 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 ...

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51 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 ...

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304 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 ...

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156 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 ...

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96 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 ...

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221 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 ...

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95 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)...

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**1**answer

154 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(...

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172 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 $...

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188 views

### Reference Request on logarithm derivative of L-functions

I'm looking for references that show almost all Dirichlet characters $\chi \mod q$ satisfy
$$|\frac{L'}{L}(1+it, \chi)|=o(\log q)$$
where $t\in \mathbb{R}$ is fixed. I have been able to adapt a method ...

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**1**answer

174 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.

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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 ...

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94 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 ...

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**2**answers

273 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 ...

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146 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{...

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60 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}...

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144 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 ...

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139 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 ...

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125 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 $g$.

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203 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 ...

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90 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 ...

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140 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 ...

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**1**answer

125 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 ...

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151 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 ...

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134 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 \...

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**4**answers

335 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 $\frac{1}{2}<\Re(s)\leq 1$ ?

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315 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 $0\leq a<b \leq 1$?

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141 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 ...

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777 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 ...

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87 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 $...

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46 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....

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415 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 ...

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**1**answer

173 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 ...