# Questions tagged [cv.complex-variables]

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

2,299
questions

**5**

votes

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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**

votes

**0**answers

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}\...

**-5**

votes

**0**answers

41 views

### 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,
$\arg(1-z^2) = \arg(z)$.
...

**-4**

votes

**0**answers

78 views

### 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(\...

**3**

votes

**0**answers

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}\...

**0**

votes

**1**answer

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**

votes

**3**answers

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 ...

**1**

vote

**1**answer

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(...

**0**

votes

**0**answers

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 \...

**1**

vote

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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 $f(x) = |g(x)|^2, x \in \mathbb{R},$ ...

**9**

votes

**1**answer

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)...

**0**

votes

**0**answers

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**

votes

**0**answers

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 ...

**0**

votes

**0**answers

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 ...

**-1**

votes

**0**answers

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 ...

**2**

votes

**1**answer

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**

votes

**1**answer

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 $F(s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s}$ ...

**0**

votes

**1**answer

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**

votes

**1**answer

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\}...

**1**

vote

**1**answer

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

**1**answer

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**

votes

**1**answer

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{...

**0**

votes

**0**answers

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 $\mathbb{S}_{\rho_1; \rho_2}$. ...

**0**

votes

**1**answer

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**

votes

**1**answer

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 \...

**0**

votes

**0**answers

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

**1**answer

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**

votes

**0**answers

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 \...

**1**

vote

**0**answers

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**

votes

**0**answers

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 ...

**0**

votes

**0**answers

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

**0**answers

161 views

### No common roots of complex polynomial and of its derivative

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

**1**

vote

**1**answer

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{...

**0**

votes

**0**answers

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 ...

**-1**

votes

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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**

votes

**0**answers

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

**1**answer

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**

votes

**0**answers

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**

votes

**0**answers

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

**1**answer

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

**1**answer

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**

votes

**1**answer

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

**0**answers

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

**0**answers

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

**2**answers

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 ...