# Questions tagged [cv.complex-variables]

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

3,137
questions

2
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0
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### Techniques of showing the Order

In their paper The integral of Riemann Xi Function, Lagarias and Montague show that the integral
$$\Xi_{\lambda}^{-1}(z)=2\int_0^{\infty}e^{\lambda u^2}\phi(u)\Big( \frac{\sin zu}{u}\Big)\;du$$
is ...

1
vote

1
answer

100
views

### Zeroes of entire function on $\mathbb C^n$

Let $n\ge 2$ be an integer and let $f$ be an entire function on $\mathbb C^n$. Let $A$ be a subset of $\mathbb R^n$ with positive $n$-dimensional Lebesgue measure. Then if $f$ vanishes at $A$, this ...

4
votes

1
answer

281
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### Conjectured closed form of $\int_0^1 \frac{\text{Li}_2\left(\frac{x}{4}\right)}{4-x}\,\log\left(\frac{1+\sqrt{1-x}}{1-\sqrt{1-x}}\right)\,dx$

After reading some meta posts, I've decided to post this question on MathOverflow since I didn't receive any comments or answers on MSE
Certainly, I apologize for any oversight. Here's a more refined ...

6
votes

1
answer

183
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### Mellin-Barnes integral representation of the exponential function with a non-real argument

I have been studying a definite integral that I found out to be a particular (and possibly one of the simplest) case(s) of the arcane Mellin-Barnes integral. Solving this problem would lead to a ...

5
votes

0
answers

173
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### Reference request: Automorphisms of $\mathbb C\{x,y\}$ which preserve the equation of the cusp, $x^3 - y^2$

In my research I encountered automorphisms of the ring of convergent power series
$$\varphi: \mathbb C\{x,y\} \to \mathbb C\{x,y\},$$
which preserve $f = x^3 - y^2$, i.e. $\varphi(f) = f$. I'm ...

1
vote

1
answer

98
views

### Problem in understanding maximum principle for subharmonic functions

I am reading subharmonic functions and their properties from the book From Holomorphic Functions to Complex Manifolds by Grauert and Fritzsche. Let me first define what a subharmonic function is.
...

4
votes

1
answer

120
views

### Upper bound for the $n$-th derivative of a rational function $\frac{f}{f+g}$

Let $f$ and $g$ be real polynomials with nonnegative coefficients. Let
$$
h = \frac{f}{f+g}.
$$
I want to prove that the $n$-th derivative of $h$ satisfies:
There exists $C > 0$ such that
$$
|h^{(...

0
votes

0
answers

125
views

### “Holomorphic” bump function

I was wondering in what sense can I construct a holomorphic “bump function”? Now, of course we cannot really construct a holomorphic bump function in the usual sense, but I have a much rougher idea in ...

3
votes

0
answers

84
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### Error function of the second moment of the divisor function

It is easy to show that the second moment of the divisor function has asymptotics:
$$\sum_{n\leq x} d_0(n)^2 = xP(\log(x))+E_2(x)$$
Where $P$ is some polynomial and that:
$$E_2 = o(x)$$
Previously, ...

-1
votes

0
answers

55
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### Linearising composition of functions [closed]

Is there a straightforward way to linearise the composition of functions? For instance, if $f$ is complex-valued and univalent (so that the notion of an inverse for $f$ makes sense) how would one go ...

0
votes

1
answer

68
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### Accessible points of a simply connected domain

We know that if $U$ is an open subset of $\mathbb{\widehat C}$ (extended complex plane), a point $v\in\partial U$ is called accessible from $U$ if there exists a curve $\gamma:[0,1)\to U$ such that $\...

1
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0
answers

179
views

### Constructing curves with large tangent space in complex variety

Suppose $M$ is a (singular) complex analytic/algebraic variety. Then for every $p\in M$ there exists a (possibly reducible) curve $C \subset U\subseteq M$ containing $p$ such that $T_pC=T_pM$, where $...

6
votes

2
answers

301
views

### Infinite sum of even Bessel functions - Identities

Recently, I came across the following identities among first-kind Bessel functions, namely
$$
2\sum_{k=1}^{\infty}(-1)^k\,k^5\,J_{2k}(x) = \frac{x^2}{4}\left[x\,J_1(x)-J_0(x)\right] \label{1}\tag{1}
$$...

2
votes

1
answer

357
views

### Limit of an infinite series with quadratic arguments

I have encountered a limiting process on some infinite series. So, I would like to ask:
QUESTION. Assume $n$ is an even positive integer. Is this true?
$$\lim_{r\rightarrow1^{-}}\sum_{j=1}^{\infty}\...

0
votes

1
answer

120
views

### Small phase approximation

Does anyone known how to prove that if $|\phi_k (r)| \ll 1$ for all $r$ and all $k=1,...,n\,$, the following equation
$$ S=\left|\int_0^\infty A(r)e^{-i[\phi_0(r)+\sum_{k=1}^n \phi_k(r)]} dr \right|^2 ...

1
vote

1
answer

94
views

### Characterizing the unimodular functions from the closed disk $\overline{\mathbb{D}}$ to $\mathbb{C}$ with constraints

Let $\mathbb{D}$ be the open disc. It is well known that if $f:\mathbb{D}\to\mathbb{C}$ is analytic, continuous on the boundary, and is unimodular (say with a finite number of zeros) then $f$ is a ...

-2
votes

0
answers

27
views

### A question about the complex function's continuity [migrated]

Suppose that $D$ is a connected open set, $n$ is a natural number. Show that: $f\in C^{n}$ if and only if $\frac{\partial^n f}{\partial z^k\partial\bar{z}^{n-k}}(0\leq k\leq n)$ is continuous on $D$.
$...

2
votes

1
answer

131
views

### Entire function of finite order with deficient value

There are some sufficient conditions for an entire function to have a finite deficient value e.g., if the order $\rho$ of an entire function $f$ is such that $2<\rho<+\infty$ with all but ...

2
votes

0
answers

157
views

### Splitting of de Rham cohomology for singular spaces

I am currently trying to wrap my head around the following splitting result by Bloom & Herrera (here is a link to the ResearchGate publication) for the de Rham cohomology of (in particular) a ...

2
votes

0
answers

132
views

### Beyond Watson's lemma

Suppose $f:[0,1]\rightarrow \mathbb{C}$ is a smooth function, which I wish to approximate near $0$. Watson's Lemma implies that I can find a smooth function $a:[0,1]\rightarrow \mathbb{C}$ such that:
$...

-1
votes

1
answer

155
views

### Best approximation of the modulus function

While there is extensive study regarding the best approximation of function with polynomial functions in the real domain, the study of approximation of complex variables becomes much sparse. See this ...

1
vote

1
answer

92
views

### Interpolation by holomorphic functions of small exponential type on a half-plane

Let $\{a_n\}_{n=1}^\infty$ be a sequence of complex numbers satisfying $|a_n|\le n^2$ and $|a_n|\to \infty$. I'm looking for a function $h(z)$ such that:
(a) $h$ is holomorphic on a half-plane $\{\Re(...

0
votes

1
answer

293
views

### Necessary conditions for convergence of convolution

In math.SE, I've asked a question about the convergence of convolution of two functions which have bilateral Laplace transform and also have disjoint Region Of Convergence (ROC) but the question didn'...

3
votes

2
answers

461
views

### Approximation for complex variables

Approximation theory, which aims to provide the optimal polynomial function approximating the target function in a given domain such as $x\in[-1,1]$, has been well-developed for real variables. In ...

2
votes

0
answers

135
views

### The Hausdorff measure of intersection of annulus and conformal curve

Recently I came across a problem in my research. Let $g:[0,1]\to\mathbb{C}$ be a restriction of a conformal map that is defined in a simply connected domain $\Omega\subseteq\mathbb{C}$ that include $[...

3
votes

0
answers

71
views

### Let $f: X \to D$ be a proper holomorphic submersion. Does a holomorphic form on $X$, closed on each fiber of $f$, have holomorphic coefficients?

Let $D$ be the unit disc in $\mathbb{C}^n$ and let $f: X \to D$ be a proper surjective holomorphic submersion, which is trivial as a smooth fiber bundle, with connected fiber $F$. We get an induced ...

6
votes

2
answers

629
views

### On the asymptotic behaviour of the series $\sum_{n=1}^{\infty} \left ( \frac{\zeta(ns)}{n}+\frac{s}{1-ns}\right )$ near $s=0$

I am interested in determining the behaviour of the the series/function
$$f(s)=\sum_{n=1}^{\infty} \left ( \frac{\zeta(ns)}{n}+\frac{s}{1-ns}\right )$$
near $s=0$. It is clear that $f(0)$ is undefined....

0
votes

0
answers

109
views

### Is any singularity a subgerm of $(\mathbb{C}^n, 0)$?

I am studying singularity theory. I have often come across, in the literature, the sentence which says "let $(X,0) \subset (\mathbb{C}^n,0)$ be a singularity". Here a singularity is a ...

9
votes

0
answers

1k
views

### How complicated can an elementary antiderivative get?

I asked this question on MSE here.
I recently learned that there are many very large numbers that have been defined, such as $\operatorname{TREE}(3)$ and many others that are too big to be written ...

1
vote

1
answer

83
views

### Perturbing pole of Laurent polynomial/series in a single summand

I am working with the ring of Laurent polynomials $\mathbb{F}[X,X^{-1}]$ over $\mathbb{F}$ for some algebraically closed field $\mathbb{F}$ of any characteristic. I encountered a problem emerging from ...

1
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0
answers

55
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### Complex geodesic coordinate, local ramified map, and the conic metric

Remark: I have asked this question in MSE, however, I got no responses. This is the reason I come to ask here. I am looking forward to some advices. Thanks in advance
Let $X$ be a compact Kaehler ...

-3
votes

2
answers

297
views

### When is $\Re(\zeta(s)) - \Im(\zeta(s)) = 0 $ with $\Re(\zeta(s))\neq 0$ and $\Im(\zeta(s))\neq 0$? [closed]

When is $\Re(\zeta(s)) - \Im(\zeta(s)) = 0 $ for $0<\Re(s)<1$. Here $\zeta$ denotes the Reimann zeta function. Does the solution live on a vertical line? Or is this another coincidence when both ...

1
vote

1
answer

127
views

### Carleman's Liouville theorem for entire functions bounded along every ray

There is a long history on constructing entire functions bounded along every direction. For example, we refer to Burckel's math review on Newman (Amer. Math. Monthly 1976 MR0387593) or this ...

3
votes

1
answer

109
views

### The number of components of the preimage of a continuum for a polynomial

Given a polynomial f with degree d, we can define a dynamical system $(\mathbb{C}, f)$. If we have a proper continuum $N \subset \mathbb{C}$, it is known that the set $f^{-1}(N)$ has at most $d$ ...

1
vote

0
answers

77
views

### Kähler potential for ALEs from resolving $\mathbb{C}^2/\mathbb{Z}_2$:

I am reading the famous paper of Kronheimer “The construction of ALE spaces as hyperkähler quotients”
I want to calculate explicitly the metric on the ALE spaces, obtained by a family of resolution of ...

0
votes

1
answer

63
views

### Recovering coefficients from certain parametric complex maps

Consider the parametric complex map $f_{A,B,v}: \ \mathbb{C}^n \rightarrow \mathbb{C}$ defined as:
$$
f_{A,B,v}(x) = Ax\cdot Bx \ |v \cdot x|^2,
$$
where $A,B$ are $n \times n$ complex matrices, $v \...

1
vote

0
answers

107
views

### Looking for examples of kernels with scalar Pick property but not the complete Pick property

I am studying Pick Interpolation and Hilbert Function Spaces by Agler and McCarthy.
A kernel $k$ on a set $X$ is said to have $M_{s,t}$ Pick property whenever $x_1,x_2, \ldots , x_n \in X$ and $W_1, ...

1
vote

0
answers

80
views

### Functional inequality with complex variables

I am interested in considering a function $C(\tau)$ of the complex variable $\tau = t + i\eta$ such that
$C(\tau)$ is analytic for $\Re(\tau)=t>t_0\ge0$
$\exists$ a constant $C_0$ and a function $...

0
votes

0
answers

79
views

### The asymptotic behaviour of the Fourier transform of a certain class of radially symmetric functions

Fix $\theta\in (-\pi/2,\pi/2)$ and let $a>0$. Suppose that $f:\mathbb{C}\rightarrow \mathbb{C}$ is analytic in $S:=\{z\in \mathbb{C}: |\arg{z}|<\pi/2\}$ and
$$|f(z)|\sim |z|^{-a},\qquad |z|\to \...

0
votes

0
answers

104
views

### Affine scheme over ring of meromorphic functions with finite poles on unit circle

I am looking into the set $S$ of meromorphic functions with a finite number of poles on the unit circle (i.e., rational functions with poles on the unit circle). I assume that any $h\in S$ has the ...

3
votes

0
answers

79
views

### Transformation of Julia set sequence emerging from meromorphic function

I consider a sequence of meromorphic functions on the Riemann sphere $f_k:\hat{\mathbb{C}} \to \hat{\mathbb{C}}$ for $k\in\mathbb{N}$ of the form
$$f_k(z)=\sum_{j=1}^{n_k}\dfrac{1}{(z-p_j)^{c_j}}$$
...

0
votes

0
answers

61
views

### Estimating a multiple integral of complex-valued function

I am trying to find an estimate of the following sum:
$$S(P)=\sum_i \int_{[0,\frac{\tau}{P}]^{n-1}} \left(\frac{\operatorname{li}(\tau)}{\tau}\right)^{n-1} \int_{[\frac{\tau}{P},1]} \frac{e(\gamma F(...

3
votes

2
answers

492
views

### Upper bound for complex integral

I am interested in obtaining a good upper bound for the absolute value of the following integral
$$
\left| \int_{0}^{\pi/3} e^{-itn} \left( 1-e^{it} \right)^{k} dt \right|,
$$
when $n>k>0$ are ...

7
votes

1
answer

451
views

### On a paper by Dimitrie Pompéiu and on one (in two parts) by Edmund Landau

To celebrate the new year and the future of mathematics (or the mathematics of future), I see no better way to ask a question stemming from my researches on power series.
The two papers the title ...

1
vote

0
answers

104
views

### Monomorphism/Isomorphism of $C_4$-tangent cones for complex varieties

Suppose that $(M,\mathcal{O}_M)$ is a reduced complex analytic space (or complex algebraic variety if you prefer). The tangent linear fiber space $TM$ associated to $M$ is defined as the analytic ...

2
votes

2
answers

245
views

### If $\inf\{b\in\mathbb{R}\mid\sum_{n=1}^{\infty}e^{-ax_n-by_n}<+\infty\}=1-a$ for all $a\in [0,1]$, does this equality hold for all $a\in\mathbb{R}$?

Let $\left\{x_n\right\}_{n=1}^{+\infty},\left\{y_n\right\}_{n=1}^{+\infty}\subset [0,+\infty)$ be two sequences of non-negative real numbers. Suppose there exist $\lambda\ge 1, c\ge 0$ such that $\...

2
votes

0
answers

77
views

### Does Kobayashi isometry map preserve complex geodesics?

Let $\gamma_1, \gamma_2$ are real geodesics in a domain $D$ and these two real geodesics are lying in the same complex geodesics, the question is, are $f\left(\gamma_1\right)$ and $f\left(\gamma_2\...

4
votes

1
answer

133
views

### Derivative of riemann map from the unit disk to a Jordan domain with non rectifiable boundary

Let $\gamma$ be Jordan curve such that for each point $p \in \gamma$, there exists a neighborhood $U$ of $p$ such that $\gamma \cap U$ is non rectifiable. Let $\phi: \mathbb{D} \to \Omega$ be a ...

5
votes

0
answers

206
views

### Is the Taylor map continuous?

(Skip to the bolded theorem below for my question, if you'd like)
Some context on asymptotic expansions and the Taylor map
In the setting of irregular singularities of meromorphic connections on the ...

0
votes

0
answers

63
views

### Conformally embedding a finite Riemann surface of genus g

Let $R$ be a compact Riemann surface of genus $g$ and let $S \subset R$ be a Riemann subsurface. Theorem B in Maskit's paper says that we can embed $S$ into a compact Riemann surface $P$ of genus $g$ ...