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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On the function $f(\sigma)=\int_{-\infty}^{\infty} | \frac{1}{(\sigma + it)\zeta(\sigma + it)}|^{2} \mathrm{d}t$

Define $$f(\sigma)=\int_{-\infty}^{\infty} \Big| \frac{1}{(\sigma + it)\zeta(\sigma + it)} \Big|^{2} \mathrm{d}t$$ where $\zeta$ denotes the Riemann zeta function and $i$ the imaginary unit. Is $f(\...
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1answer
125 views

Injectivity of analytic functions

Suppose $f : \mathbb{R} \rightarrow \mathbb{R}^n$ is a real analytic function on $(a, \infty)$. I have two questions: Suppose $||f(x)|| \rightarrow \infty$ as $x \rightarrow \infty$. I know without ...
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30 views

Finding all possible set of functions

Let $\{ h_n(x)\}_{n=1,..,N}$ a set of $2\pi$ periodic functions such that they satisfy the reflection property \begin{equation} e^{h_n (x+\pi) + i\bar{h}_n (x+\pi)} = \sum_m C_{nm} e^{h_m (x) + i \...
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86 views

General class of functions satisfying growth condition on a given functional

The question is inspired by Abel Plana summation formula : Is there a general class of functions $f$ that are positive valued on the positive real axis, and which satisfy the following $$\int_0^\infty ...
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1answer
138 views

Existence of entire function that yields periodicity

I have the following question: Does there exist an entire function $f(z)$ where $z=x+iy$ such that $$g(x,y) =e^{-2\pi y^2}f(z)$$ is periodic in both $x$ and $y$ direction, i.e. $$\forall x,y: g(1,y)=g(...
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1answer
209 views

Functions $f \geq 0$ on $\mathbb{R}$ which are of the form $f = |g|^2$ for some entire function $g$

I think the answer to this question must be well known. Is it possible to characterize those functions $f \colon \mathbb{R} \to \mathbb{R}_+$ which are of the form $f(x) = |g(x)|^2, x \in \mathbb{R},$ ...
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1answer
281 views

Solving equation of matrix valued functions

Given $n\times n$ matrices with entire functions entries (holomorphic on all of the complex plane $\mathbb{C}$) $A(z)=[a_{ij}(z)],B(z)=[b_{ij}(z)]$, i.e., $a_{ij}(z),b_{ij}(z)$ are entire functions ...
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68 views

Existence of a harmonic function on the upper half plane unbounded on all points of the x-axis

Does there exists a harmonic function on the upper half plane which diverges at all points of the boundary x-axis? Answers in Extension of harmonic function at infinity seem relevant but I am not sure ...
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1answer
109 views

Analytic continuation over boundaries

In D.J Newman's paper A simple analytic proof of the prime number theorem there is the following theorem: Suppose $|a_n|<1$ and form the Dirichlet series $F(s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s}$ ...
2
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1answer
261 views

Relationship between volume and area

Let $\mu(z) dV_n$ be a measure in $\mathbb{C} ^n$. Let $B_n(r) := \{z \mid \|z\| < r\}$ be the ball of radius $r$ in $\mathbb C^n$, and $\partial B_n(r) $ be the corresponding sphere. In $\mathbb{C}...
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46 views

The role of a combination of Eneström-Kakeya and Gauss-Lucas theorems: reference request or soft question, asking for this combination as tool

In past days I was trying to create problems or direct applications invoking Eneström—Kakeya and Gauss-Lucas theorems for certain arithmetic functions that I know from analytic number theory. These ...
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1answer
103 views

Analyze a function defined in terms of an integral

Here is a question that really has puzzled me for quite a while. I happened to see this function defined in terms of an integral $$f(x):=\int_0^{\pi/2}\frac{2e^{x+e^x\cos y}}{1+\left(e^{e^x\cos y}\...
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Theorem 5.3 ([Okounkov-01]) in Borodin and Gorin's lecture note

In this lecture note: https://arxiv.org/pdf/1212.3351.pdf, Theorem 5.3(P28): Suppose that the $\lambda \in \mathbb{Y}$ is distributed according to the Schur measure $\mathbb{S}_{\rho_1; \rho_2}$. ...
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1answer
90 views

Conformal mapping between two right-angled triangles

I want to derive a conformal mapping $f\!:\!A\!\to\! B$ where $A=\{ (x,y)\ |\ x\!\in\![0,1]\ \text{ and }\ 0 \leq y \leq x \}$ and $B=\{ (x,y)\ |\ x\!\in\![0,1]\ \text{ and }\ 0 \leq y \leq \frac{...
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1answer
138 views

Binomial transform of Dirichlet series

Let $\Theta(s)$ be a Dirichlet series , and let $\beta$ be its abscissa of convergence: $$\Theta(s)=\sum_{n=1}^{\infty}\frac{\theta(n)}{n^{s}}\;\;\;\;\;\;\Re(s)>\beta$$ And let $\left\{a_{n}\right\}...
2
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1answer
93 views

3D similarities and quaternions?

As is well-known, in dimension 2, a linear map $f : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ is a direct similarity if, once we identify $\mathbb{R}^2$ with $\mathbb{C}$, $f$ is of the form $$\forall z \...
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69 views

Meromorphic functions on a modular curves of genus $0$ that take each value exactly once

Let $\Gamma$ be a congruence subgroup of $\operatorname{SL}_2(\mathbb Z)$, and let $\mathfrak H$ be the upper half-plane. Let $X(\Gamma)$ be the compactification of $\Gamma\backslash\mathfrak H$. Then ...
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1answer
141 views

Continuous extensions of Riemann mappings

Let $K$ be a compact set in $\mathbb C$ without interior. Suppose, additionally, that $K$ is a retract (or equivalently $K$ connected, $K$ locally connected and $\mathbb C\setminus K$ connected). ...
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72 views

Estimates for tensors using local coordinates

Suppose that we have a Kähler manifold $(M, \omega_0)$ and another $(1,1)$ form $\eta$ on $M$. Let $\varphi$ be a smooth function such that $\omega_{\varphi} = \omega_0 + i \partial \bar \partial \...
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163 views

Understanding Krantz's proof of Hefer's lemma in $\mathbb{C}^2$

Note: I initially phrased the question in a different way, and it did not receive much attention. In the hope to make it more interesting, I have included a (long) introduction to contextualize and ...
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Bounding the absolute value of a complex integral with itself

I already asked a similar question on this topic, but after a small discussion, I noted that I did must boil down the problem such that the solution space so to say to maybe have a concrete answer. I ...
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75 views

injective holomorphic mapping between unit disk and unit polydisk

In $\mathbb{C}^n,\ n\geq 2$, there is no bijection between unit disk $B^n(0,1)$ and unit polydisk $P^n(0,1)$. But if we wish to find injective holomorphic mapping from unit disk to polydisk(whose ...
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62 views

Image of transcendental meromorphic functions

Let $f$ be a trancendental meromorphic function such that $f'(z) \ne 0$ for all $z \in \mathbb{C}$. Let $\Pi$ be the stereoprojection map from the north pole on the unit sphere. My question is the ...
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1answer
84 views

Roots for $p(w)=n+\sum_{j=1}^{m}\frac{v_{j}}{w-v_{j}}$

Let $v_{j}\in \mathbb{C}, 1\leq j\leq m$ and $w\in \mathbb{C}\setminus \{v_{j}\}_{j=1}^{m}$ and $n>0$. Q: Can we say anything about the m roots $w_{1},...,w_{m}$ of $$p(w)=n+\sum_{j=1}^{m}\frac{...
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161 views

No common roots of complex polynomial and of its derivative

Our specific context Here is our specific contour integral $$\int_{\Gamma_{0}}F\big(\sum_{w:p_{z}(w)=0}\frac{1}{w^{a}}\frac{1}{n+\sum_{j=1}^{m}\frac{v_{j}}{w-v_{j}}} \big)\frac{dz}{z},$$ ...
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51 views

Bounding the absolute value of a complex integral

I'm working on some problems involving Fourier transforms and convolution problems and there is one problem I cannot solve. In my situation we have a complex valued function $f(ix)$, with $x\in\mathbb{...
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1answer
149 views

Cauchy's Integral with quadratic exponential term

As I was studying the Cauchy's integral formula, I tried to do the integral: \begin{equation} I = \int\limits_{-\infty}^{\infty} \frac{1}{x - a} e^{(i A x^2 + i B x)} dx \end{equation} with $A>0, ...
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96 views

Analytic and algebraic definitions of intersection multiplicity of two complex algebraic curves coincide

There are two definitions of intersection multiplicity of two complex algebraic curves. One is given in An introduction to algebraic curves by Griffiths. Let $t \mapsto (t^k, y(t^k) )$ be the local ...
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41 views

Oscillatory integral independent of a parameter

Let $h: \mathbb{R} \mapsto \mathbb{C}$ be a positive definite function, continuous at the origin. (In fact, $h$ is the Fourier transform of a finite measure). Define the oscillatory integral $$Q(t) :=...
4
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1answer
88 views

minimizing weighted length of closed curves

Let $\mathcal{A}$ be the family of closed smooth curves in the right half of the complex plane $\mathbb{C}$ such that any curve in the family must enlose the point $z=1$ and tangent to the $y$-axis at ...
3
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1answer
97 views

Reference request: The transform of a bounded random variable has a zero in the complex plane

Together with coauthors I'm working on a paper where we use the following Proposition: If a real-valued random variable $X$ has bounded support, then except in the trivial case that $X$ has all ...
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1answer
418 views

Partial sums of $\sum_0^\infty z^n$

Let $z$ be a complex number with $|z|<1$. For every subset $A\subset\mathbb N$, the series $\sum_{m\in A}z^m$ is convergent. Denote $S(A)\in\mathbb{C}$ its sum and $\Sigma_z$ the set of all numbers ...
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80 views

Equivariant resolution of singularities with equivariant centres

From what I understand, given a complex projective variety X inside a compact complex manifold Y, according to Hironaka, there is a sequence of $r$ blowups $Y_i$ of Y along complex submanifolds (...
3
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168 views

Stokes's Theorem with singularities on projective line

Let $X$ be a complex manifold and $\omega\in \Omega(X\times \mathbb{P}^1)$ a form. I met the following identity: $$\int_{\mathbb{P}^1}(\partial_z\bar{\partial}_z\omega) \log|z|^2=\int_{\mathbb{P}^1}\...
4
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1answer
453 views

A problem on polynomials

Let $P(z)$ be a polynomial of degree $n$ with $|P(z)|\leq 1$ on $|z|=1$ and $P_m(z)$ be a partial sum of $P(z).$ How large $P_m(z)$ can be on $|z|=1?$
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54 views

Holomorphic semigroups vs analytic semigroups

Is there any difference between the two notions in the theory of semigroups? In the literature, we find some monographs use the farmer while others use the latter. I expect that they are always the ...
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118 views

Derivative of a polynomial $P(z)$ and the derivative of the conjugate reciprocal of $P(z)$

Let $P(z)$ be a polynomial of degree $n$ having no zeros in $|z|<1.$ Let $Q(z)=z^n\overline{P(1/\overline{z})}.$ Then it is an easy exercise to show that $\Re\left(zP'(z)/P(z)\right)= \sum_{k=1}^n\...
2
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0answers
75 views

Theta-function in the lower half-plane

Standard theta function $$\vartheta(q)=\sum_{n=-\infty}^\infty q^{n^2} \qquad\qquad(1)$$ has a natural boundary of analyticity at $|q|=1$. This means that it can not be used to regularize expressions ...
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27 views

Fixing coefficients using analytic structure

I am trying to understand an exercise in a set of lecture notes on random matrices - Eynard - Random matrices - given in the paragraph following (4.6.60) (pp. 70–71). Specifically, I am trying to fix ...
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1answer
84 views

Logarithms of $L$-functions of irreducible characters of Galois group

We know that the $L$ functions of Dirichlet characters $\chi$ of $(\mathbb Z / m\mathbb Z)^\times$ satisfy the property that $\log L(s, \chi)$ is holomorphic for $\Re(s) \geq 1$ if $\chi$ is a ...
2
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1answer
86 views

Existence of analytic continuation of Dirichlet series corresponding to the indicator sequence of a complement of a special multiplicative set

Let $K/ \mathbb Q $ be a finite Galois extension and let $X$ be a proper non-empty subset of the Galois group $G=Gal(K/ \mathbb Q)$ that is closed under conjugation. Consider a set of integer primes $...
3
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1answer
78 views

A question on preimage of a locally injective meromorphic function

Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a meromorphic function such that $f'(z) \ne 0$ for all $z \in \mathbb{C}$. Let $\Gamma \subset \mathbb{C}$ be a curve which has no self intersections. If ...
3
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1answer
193 views

To show holomorphicity of a certain infinite series of functions

I have the following sequence of holomorphic functions on $(f_n(s))_{n \geq 1}$ on the closed region $R:= \{ s \in \mathbb C : \Re(s) \geq 1\}$ by $$f_n(s) := \begin{cases} \log(1+p^{-s})-p^{-s}\...
2
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0answers
211 views

On the logarithmic derivative of an analytic function

I have reached upto this situation while exploring some properties of analytic functions which are bounded on the unit circle. Suppose $f(z)$ be such an analytic function in the closed unit disc ...
4
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1answer
209 views

Fun examples relating to Hopf surfaces

A Hopf surface is a compact complex surface whose universal cover is complex analytically isomorphic to $\mathbb{C}^2 \setminus \{ 0 \}$. I would like to know whether anyone has any of the following ...
4
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1answer
116 views

Distribution boundary value of analytic function and wave front sets

Assume $f(z)$ is analytic in the tube domain $\mathbb R^n\oplus iC$, where $C\subset \mathbb R^n$ is a convex cone. Under the assumption $|f(x+iy)|\leq 1/|y|^k$, we know by a Theorem of Martineau (see ...
10
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1answer
314 views

Sendov's conjecture

It has been more than fifty years for famous Sendov's conjecture which states that if $p(z)$ is a polynomial of degree $n$ having all its zeros in the unit disc $|z|\leq 1$ then each of the n ...
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58 views

zero extension of positive currents are always positive

In Demailly's Complex Analytic and Differential Geometry page 139: He said the trivial (zero) extension of the positive current $T$ (on $X\setminus E$), which denoted by $\tilde T$ is always positive ...
6
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1answer
113 views

The state of art of the singular Levi problem — and hyperkähler quotients

One of the versions of the classical Levi problem asks the following: Let $X$ be a complex manifold. Is it true that $X$ is Stein iff $X$ admits a smooth exhaustion strictly plurisubharmonic ...
4
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104 views

Resources for divergent / asymptotic series

This series is divergent; therefore, we may be able to do something with it. -- Oliver Heaviside Other than the usual references given in Wikipedia and Mathworld, which resources have you found ...

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