# 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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votes

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80 views

### Do analytic functionals form a cosheaf?

Let $X$ be a complex-analytic manifold.
Consider the sheaf of holomorphic functions $\mathcal{O}_X$ as a sheaf with values in the category of locally convex vector spaces. For $U\subseteq X$ open, we ...

**-5**

votes

**0**answers

42 views

### What is the Laurent series of z+(1/z)? [on hold]

What is the Laurent series of z+(1/z)? Is it just the series itself?

**3**

votes

**0**answers

53 views

### Metric with singularities on Riemann Surfaces and the associated Laplacians

I have asked this question on Math Stack Exchange
Metric with singularities and associated Laplacian
but I have not got any answers/comments, therefore I post this question on the MO.
Suppose $M$ ...

**0**

votes

**0**answers

30 views

### Is there a way to categorise the valleys of a holomorphic function (potentially of $\geq 2$ variables) (multidimensional steepest descent)

More specifically, I am particularly interested in the question: given some $f:\mathbb{C}^n \to \mathbb{C}$, can we categorise $\mathbb{C}^n$ by which valley the steepest descent curves of a point (...

**5**

votes

**1**answer

184 views

### Dependence of a solution of a linear ODE on parameter

Is the following theorem known, or can be easily derived from known results?
Consider the differential equation
$$w''-kz^{-1}w'=(\lambda+\phi(z))w,$$
where $k>0$ is fixed, $\lambda$ is a large (...

**2**

votes

**0**answers

106 views

### Is there any neat way to calculate the Fourier transform for an inverse Vandermonde determinant?

$$\mathrm{PV}\int_{\mathbb R^n} \frac{e^{-i\langle w, x\rangle}}{\prod_{j<k}(x_k-x_j)}dx=?$$
Other than integrate this term by term (which might look crazy)?
Let $f(x)=1/\prod_{j<k}(x_k-x_j)$, ...

**3**

votes

**2**answers

360 views

### Reference request: Oldest complex analysis books with (unsolved) exercises?

Per the title, what are some of the oldest complex analysis books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there. I am aware of the ...

**1**

vote

**1**answer

116 views

### An equation with Gamma Euler function in critical strip

Let
$$
D=\{z \in \mathbb{C} : 0<\Re(z)<\frac{1}{2} \text{ or } \frac{1}{2}<\Re(z)<1 \}
$$
that is the critical strip without critical line.
I have to find if the following equation, with ...

**3**

votes

**0**answers

103 views

### Injective resolution of the ring of entire functions

Let $R$ be the ring of entire functions on $\mathbb{C}$. I heard that the concrete value of the global dimension of the ring depends on continuum hypothesis.
I would think that the injective dimension ...

**3**

votes

**1**answer

48 views

### Rational approximation on rotation invariant compact subsets of complex plane

What does the Vitushkin's theorem say about the equality $A(K) = R(K)$ in the special case when $K$ is rotation invariant? More precisely, what are necessary and/or sufficient conditions on $\{|k|: k \...

**2**

votes

**2**answers

104 views

### Coefficients of entire functions with specified zero set

Let $Z \subseteq \mathbb{C}$ without limit point. By the Weierstrass factorization theorem there is an entire function $h$ those zero set is $Z$. Let $a_n > 0$ be a sequence where $\lim_n \sqrt[n]{...

**3**

votes

**1**answer

156 views

### Local phase statistics of the nontrivial Riemann zeros

(The question is inspired by Owen Maresh's post)
The local phase of a nontrivial zero $s$ of the Riemann $\zeta$ is the argument of $\zeta'(s)$.
Numerical results on the first 10000 zeros suggest ...

**1**

vote

**0**answers

101 views

### Defining integrals by residue theorem

I have always been interested in alternative definitions of mathematical objects. I wonder if one can craft an useful definition of definite integral by using the Residue Theorem from complex analysis....

**0**

votes

**0**answers

141 views

### An (unusual) uniqueness theorem for analytic functions

Let $L$ be the class of analytic functions in $\mathbf{C^*}$
with positive Laurent
coefficients:
$$f(z)=\sum_{-\infty}^\infty c_nz^n,\quad 0<|z|<\infty,\quad c_n\geq 0.$$
Each $f\in L$ has a ...

**0**

votes

**1**answer

74 views

### Solving or bounding the real part of the integral $\int_0^{2 \pi i m} \frac{e^{-t}}{t-a} dt$

I would be interested in finding a closed form or, at least, bounding (in terms of $m$ as it becomes larger) the real part of the following itnegral:
$$f(m,a):=\int_0^{2 \pi i m} \frac{e^{-t}}{t-a} ...

**3**

votes

**0**answers

37 views

### Modulus of image of a curve family in a rectangle

I don't expect to get a positive answer to this question but I may as well try.
Let $R$ be the rectangle in $\mathbb{C}$ given by $\{z=x+iy: 0\leq x \leq l, 0 \leq y \leq h\}$ for some $l,h>0$. ...

**0**

votes

**1**answer

69 views

### meromorphic extension of dirichlet series

Suppose $\{a(n)\}_{n\ge 1}$ is a bounded complex sequence. Let $\phi(s)=\sum_{n\ge 1} \frac{a(n)}{n^s}$. Obviously, the Dirichlet series $\phi(s)$ is absolutely convergent for $\mathcal{R}(s)>1$. I ...

**2**

votes

**1**answer

119 views

### Must $q$ be analytic?

I have a continuous function $q:\mathbb{R}^+ \to \mathbb{R}^+$. An interesting property of this function is that
$$F(s) = \frac{e^{-q(s)}q(s+1)}{1-e^{s-q(s)}}$$
which also takes $\mathbb{R}^+ \to \...

**2**

votes

**1**answer

142 views

### Differences of $\omega$-plurisubharmonic functions

Let $X$ be a complex manifold, and $\omega$ a Kähler form on $X$.
A smooth function $\phi$ on $X$ is $\omega$-plurisubharmonic ($\omega$-psh for short) if the form $\omega+\sqrt{-1}\partial\bar{\...

**5**

votes

**0**answers

158 views

### Regular functions vs holomorphic functions

Let $X$ be an affine smooth variety over the complex numbers, $X^{an}$ its associated smooth complex analytic space, and $\mathcal{O}$, resp. $\mathcal{O}^{an}$ the respective structure sheaves.
Is ...

**4**

votes

**1**answer

68 views

### Existence of Laurent series with zeroes at $𝑒^2𝑛$ (𝑛∈ℕ0 ) and even faster coefficient decay

This is an extension of an earlier question of mine which corresponds to the case $A = 1$. Precisely, my question is as follows:
Given $A > 0$ fixed but arbitrary, is there a non-trivial ...

**4**

votes

**1**answer

160 views

### Zeros of derivatives of Dirichlet Eta function

Let
$$
\eta^{(d)}(z) =
\sum_{n=1}^\infty
\dfrac
{(-1)^d(-1)^{n-1}\ln(n)^d}
{n^z}
$$
be the derivative of Dirichlet Eta function of order $d$.
Does it exist any known or not known zero of $\eta^{(d)}...

**0**

votes

**1**answer

106 views

### What are the holomorphic automorphism groups of unit ball, polydisc, and Hartogs domain in C^n ( n>1)? [closed]

What are the holomorphic automorphism groups of unit ball, polydisc, and Hartogs domain in $\mathbb{C}^n$ ($n>1$) ? I would be pleased if you tell me.

**9**

votes

**1**answer

147 views

### Existence of Laurent series with zeroes at $e^{2n}$ ($n \in \Bbb{N}_0$) and extremely fast coefficient decay

I am working on a problem in harmonic analysis, which I converted into the following existence problem concerning Laurent series. I am a bit at a loss concerning this problem, since my knowledge of ...

**10**

votes

**0**answers

275 views

### Riemann–Hilbert-type problem

Let $P$ be a fixed pentagon in the hyperbolic plane $\mathbb H^2$ with all the angles equal to $\pi / 3$. Let $w_1, w_2, \dots, w_5$ be the sides
of $P$ going in the counterclockwise order. We are ...

**3**

votes

**1**answer

95 views

### Decay of the binomial expansion of $f^{\circ k}$

Suppose $f$ is a holomorphic function in a neighborhood of zero fixing zero. Suppose $f'(0) = \lambda$ and $0<\lambda < 1$. It's not so hard to prove that $f^{\circ k}(z) = f(f(\ldots\text{$k$ ...

**3**

votes

**0**answers

36 views

### Extremal metric for image of a curve family

Let $U\subset \mathbb{C}$ be a domain and $\Gamma$ some family of curves in $U$ with $\textrm{mod}(\Gamma)<\infty$ and such that $\rho$ is an extremal metric for the modulus. Suppose we are given a ...

**27**

votes

**2**answers

1k views

### A sum involving roots of unity

Let $n$ be a positive integer and $\zeta$ be a primitive $n$th root of unity. It is not hard to show that
\begin{align*}
\sum_{k=1}^{n-1}\frac{\zeta^k}{1-\zeta^k}=\frac{1-n}{2}.
\end{align*}
Since $\...

**1**

vote

**0**answers

86 views

### Fourier inversion formula for compactly supported distributions

I know that the Fourier transform of a compactly support distribution $u\in \mathscr{E}'(\mathbb{R}^{n})$ is smooth and also satisfies
$$
|\hat{u}(\xi)|\leqslant C_{N}(1+|\xi|)^N,\label{1}\tag{1}
$$
...

**0**

votes

**0**answers

84 views

### A certain ratio condition for polynomials with real coefficients

Let $p:\mathbb{C} \longrightarrow \mathbb{C}$ be a polynomial with real coefficients and suppose that $p$ satisfies
\begin{equation}
\frac{p(y)}{y} \le \frac{p(x)}{x} \tag{*} \label{ratcond}
\end{...

**0**

votes

**1**answer

81 views

### Generalized Lambert W Function

I am looking for inverse functions for the following family of functions:
$
\begin{aligned}
f_0(z) &= z+e^z \\
f_1(z) &= ze^z \\
f_2(z) &= z^z \\
&\cdots \\
f_{n+1}(z) &=...

**5**

votes

**1**answer

111 views

### Failure of Borel-Schmid quasi-unipotency theorem in non-algebraic case

Let $\mathcal{X} \to \Delta^{\times}$ be a smooth family of compact Kähler manifolds over a punctured disc.
When this family is algebraic, the celebrated quasi-unipotency theorem (I think, due to ...

**2**

votes

**0**answers

64 views

### How to compute expansion factors for hyperbolic rational maps?

It is a commonly-referenced result about certain rational maps acting on $\mathbb{\hat{C}}$ that they are expanding on a neighborhood of their Julia sets. A sufficient condition to be expanding is ...

**0**

votes

**2**answers

150 views

### Dominant root of a family of polynomials

Let $f(x)=x^5-x^4-x^3-x^2-x-c$, where $c>2$ is a real number. It is easy to prove that there exists a positive real root $\alpha>2$ of $f(x)$ and all the other roots are non real.
Also, by ...

**2**

votes

**1**answer

93 views

### Modulus bounded by Nevanlinna characteristic in several variables

Let $f:\mathbb{C}^n\to\mathbb{C}$ be an entire holomorphic function of $n$ complex variables. Then its Nevannlinna characteristic equals
$$
m_f(r)=\int_{\partial B(r)}\log^+|f(z)|d\eta(z),\quad\forall ...

**4**

votes

**0**answers

163 views

### When is entire function bounded on a ray?

Let $f(z)=\sum_{n=1}^\infty c_n z^n$ be entire function on the complex plane. May we express the property $\int_0^\infty |f(x)|^2 /x dx<\infty$ or some other property controlling the behavior for ...

**4**

votes

**1**answer

170 views

### On the roots of Bernoulli polynomials

Consider the Bernoulli polynomials denoted by $B_n(z)$. Now, start plotting the set of all (combined) complex roots $\mathcal{A}_N$ of $B_n(z)$, say for $n=1,2,\dots,N$ for some enough large $N$. It ...

**0**

votes

**1**answer

91 views

### Compatible solution of PDE

Let $c=c(z, \bar z)$ be a complex function satisfying $\partial_{z} \bar c=\partial_{\bar z} c$. It follows that there exists a real function $f$ such that $\partial_{\bar z} f=-c$. Would it be ...

**12**

votes

**1**answer

665 views

### Zeros of an infinite series

Let $\sum_{j=1}^{\infty}a_{j}$ be a convergent series of positive numbers and $\{z_{j}\}_{j=1}^\infty$ a closed discrete subset of the open unit disc $\mathbb{D}$. Then $h(z):=\sum_{j=1}^{\infty}\frac{...

**3**

votes

**0**answers

130 views

### Chern number of projection-Topological magic in physics

I enclosed a computation from a well-known paper in the field of mathematical physics where the Chern number of the first Landau level is computed (the result claimed is $-1$) and the full paper can ...

**2**

votes

**0**answers

45 views

### Entire analytic functions with entire analytic Fourier transform, and corresponding distributions

I'm interested in the Fourier transform on a space of distributions that includes more than the usual tempered distributions, and in particular allows for $\delta$-distributions supported at complex ...

**4**

votes

**2**answers

145 views

### Density of Lacunary Functions

I'm curious whether lacunary functions (functions in the complex plane that are holomorphic in some open ball about the origin, but cannot be analytically extended past that ball) are typical or the ...

**0**

votes

**1**answer

68 views

### Bringing a Heun equation into canonical form

It is a well known fact that any second order Fuchsian differential equation on the complex plane $$u''(x) + p(x)u'(x) + q(x)u(x)=0$$ with exactly $4$ regular singular points may be suitably ...

**5**

votes

**3**answers

199 views

### Origin of term Ahlfors-David regular

Much of the literature on analysis in metric spaces makes use of an assumption called Ahlfors regularity or Ahlfors-David regularity. Let $q>0$. A metric space $(X,d)$ is Ahlfors(-David) $q$-...

**4**

votes

**1**answer

131 views

### Compilation of representations of holomorphic functions

Holomorphic functions are my muse. As my muse, I love drawing them different ways. Allow me to frame this as though an artist talking about his muse.
A holomorphic function $f$ on the unit disk $\...

**2**

votes

**1**answer

106 views

### Extended Abel-Jacobi theorem

Let $X$ be a compact Riemann surface, $u$ a meromorphic function on $X$ with divisor supported on a set of points $\{P_1, ..., P_n\}$, and $f$ a meromorphic function on $X$ such that $f$ has no pole ...

**3**

votes

**2**answers

264 views

### On finite extensions of the field of meromorphic functions

Let $\mathcal{M}$ be the field of meromorphic functions of one (complex) variable and $w = w(z)$ an analytic function satisfying a polynomial equation
$P(w; z) := w^n + a_{n-1}(z) w^{n-1} + \cdots + ...

**-1**

votes

**1**answer

212 views

### On a certain representation of the Riemann zeta function

Let $\zeta$ denote the Riemann zeta function. In this answer: https://mathoverflow.net/a/314066/133634, @Paul Garret considers the representation
$$\frac{\zeta(s)}{s} = \int_1^\infty (\sum_{1 \le n \...

**4**

votes

**0**answers

75 views

### LlogL 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

567 views

### An integral involving the argument of the Gamma function and the Riemann Hypothesis

Evaluate $$I=\int_{0}^{\infty} \frac{t\arg
\Gamma(\frac{1}{4}+\frac{it}{2})}{(\frac{1}{4}+t^2)^2}\mathrm{d}t$$
where $\Gamma(s)=\int_{0}^{\infty}e^{-x}x^{s-1}\mathrm{d}x.$
Note that $I$ converges ...