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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5
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0answers
425 views

On properties on a certain functional

Consider the following function: $$F(z) = \omega(z)\sin^2\left(\frac{c\Gamma(z)}{z}\right)$$ Here, $\omega(z)$ is a weight we have to construct and $c$ is a constant. The following three conditions ...
4
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0answers
46 views

Independent inner functions on the unit disk

This problem cropped up in a paper that I am writing and I have thought about it for too long to no avail: let $\mathbb{D}$ be the open unit disk in the complex plane and suppose $\varphi:\mathbb{D}\...
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Argument of 1-z^2 [closed]

Let $z=re^{i\theta}$. I am trying to find $\arg(1-z^2)$?? After some work, I derived the following: $1-z^2 = 1+\lvert z\rvert^2 (1-2\cos(\theta)) e^{i\theta}$. This implies, $\arg(1-z^2) = \arg(z)$. ...
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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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0answers
169 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}\...
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1answer
136 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 ...
6
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3answers
302 views

about the Hausdorff dimension of Removable singularities of PDE

There are some interesting phenomenons about removable singularities (or extension problems). In the theory of functions of several complex variables, we know the classical Hartogs theorem: Let f ...
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1answer
143 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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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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1answer
159 views

Locus of roots of all convex combinations of two monic polynomials, II

This post contains a revised conjecture to a conjecture I posed previously which was shown to be false. Let $p, q \in \mathbb{C}[t]$ be two monic polynomials of degree $n \ge 1$. For $\alpha \in [0,1]$...
4
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1answer
285 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 ...
7
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1answer
213 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
475 views

Complex manifold with boundary

My question is of local nature. Let $$f:\mathbb C^n\to\mathbb R$$ be a $C^\infty$ function that vanishes at $0\in \mathbb C^n$, with non-zero derivative. Then, around $0\in \mathbb C^n$, $$M:=f^{-1}(0)...
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0answers
165 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 ...
8
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256 views

How to interpret Gauss's late fragments on conformal mapping of the interior of an ellipse (to the unit disk) in modern mathematical terms?

My question refers to some not very well known (and unpublished) fragments of Gauss that treat the problem of finding a conformal mapping (angle-preserving mapping) in the complex plane from the ...
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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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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
264 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}...
2
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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 $F(s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s}$ ...
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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
139 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\}...
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1answer
104 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}\...
2
votes
1answer
165 views

Finite generation of certain graded sequences of ideals

Let $U\subset\mathbb{C}^n$ be an open set containing the origin $o$ and $Y\subset U$ a complex analytic subvariety of pure codimension $c$ with ideal sheaf $\mathcal{I}_Y$. Let $\frak{a}_{\bullet}=\{{\...
3
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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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0answers
65 views

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
151 views

Are anti-linear maps/semi-linear, such as conjugations, linear in other almost complex structures?

I have asked this on mse, but I did not get any responses even after a bounty. I started studying the book of Daniel Huybrechts, Complex Geometry An Introduction. I tried studying backwards as much ...
2
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1answer
95 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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0answers
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 ...
2
votes
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). ...
2
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0answers
73 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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0answers
92 views

Characterization of turning points for the Ramanujan's zeta function in the spirit of a definition by Arias de Reyna and van de Lune

In [1] the authors provided a definition and characterization of turning points for the Riemann's zeta function. In this post I denote the Ramanujan's zeta function as $$\varphi(s)=\sum_{n=1}^\infty\...
4
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0answers
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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0answers
35 views

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 ...
3
votes
0answers
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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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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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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1answer
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 ...
0
votes
0answers
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{...
5
votes
1answer
269 views

$L\log L$ and Hardy space on the upper half plane

Set $\mathbb{T}$ the unit circle, $dm$ the Lebesgue measure on $\mathbb{T}$ and $\mathbb{C}^+=\left\{z\in \mathbb{C},s.t.\,\Im(z)>0 \right\}$ the upper half plane. It is well-known that the Cauchy ...
1
vote
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, ...
3
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0answers
98 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 ...
4
votes
1answer
89 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 ...
0
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0answers
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) :=...
2
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0answers
119 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\...
3
votes
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 ...
4
votes
1answer
454 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 $|z|=1?$
15
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1answer
419 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 ...
2
votes
0answers
81 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 (...
2
votes
0answers
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 ...
36
votes
2answers
2k views

Computing self-intersections with complex analysis

It is possible to find the winding number of a path $C \subset \mathbb{C}$ using complex analysis: $$n = \oint_C\frac{dz}{z}.$$ You can also count the number of roots of $f(z) = 0$ inside a close ...

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