Questions tagged [cv.complex-variables]
Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
3,146
questions
14
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0
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198
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Proofs of the valence formula that avoid tricky contours?
$\DeclareMathOperator\ord{ord}\DeclareMathOperator\Im{Im}$The valence formula for a modular form asserts that if $f: \mathbf{H} \to \mathbf{C}$ is a modular form of weight $k$ on the upper half-plane $...
-2
votes
2
answers
151
views
Bounds for analytic circles
It is known that for certain particular entire functions $f(s)$ of first order, in the circle $|s| = p$, if $\epsilon$ is a positive number as small as desired, the following bound holds:
$$|f(s)| = O(...
0
votes
0
answers
46
views
Examining the Hilbert transform of functions over the positive real line
$\DeclareMathOperator\supp{supp}$Let $H:L^{2}(\mathbb{R})\to L^{2}(\mathbb{R})$ be the Hilbert transform. Let suppose we have a compaclty supported function $f \in L^{2}(\mathbb{R})$ such that $\supp(...
0
votes
0
answers
15
views
Recovering the population eigenvalue distribution from the Marchenko-Pastur distribution
Question: If we know the value of $y>0$ and the Marchenko-Pastur distribution $\nu$ (and thus also $m_\nu$), can we reconstruct the distribution $H$ from equality (1) below?
Background on the ...
4
votes
2
answers
331
views
Nontrivial invariant transformations for heat equations
It is well known that if $u$ is a harmonic function on $\mathbb R^2$ then its Kelvin transform defined by
$$ v(r,\theta) = u(\frac{1}{r},\theta)$$
is also harmonic for $r>0$. Note that the Kelvin ...
5
votes
1
answer
350
views
Lindelöf hypotheses for derivatives of zeta
The Lindelöf hypothesis says that if we have:
$$\zeta(\sigma+iT)=\mathcal O(T^a)$$
Then if one considers $\sigma=1/2$ then $\inf a=0$. Further, from convexity and the functional equation this implies ...
8
votes
1
answer
258
views
A real-valued analogue of the Weierstrass $\wp$ Function
I am interested in the following function:
$$\mathcal{Q}(z) = \sum_{w \in L^*} \frac{1}{|z-w|^2} - \frac{1}{|w|^2} \, . $$
This function is analogous to the Weierstrass $\wp$ function, the only ...
1
vote
0
answers
39
views
Looking at a frequency reassignment rule as a Möbius transform
Suppose we have some Schwartz function $h$. Denote its Fourier transform $\widehat{h}$. Let $\xi_0$, $a$, $\Delta$ be positive and fixed.
I have a function $\Omega: \mathbb{R}\times \mathbb{R}^+ \to \...
2
votes
1
answer
93
views
Bound for the $n$-th derivative of a proper rational function with no poles on the right half-plane
Suppose that
$f$ and $g$ are polynomials with nonnegative coefficients,
the degree of $g$ is greater than the degree of $f$,
$g + f$ have no zeros on the right half plane $\mathbb{C}_+ = \{z \in \...
6
votes
0
answers
564
views
Generating functions in countable commutative monoids
Let $f: \mathbb{N}_0 \rightarrow \mathbb{C}$ be a function. The power series of $f$ can be viewed as the function $\mathscr{P}_f : q \mapsto \sum_{n \in \mathbb{N}_0}^{} f(n)q^n$ where $q \in \mathbb{...
2
votes
0
answers
114
views
Techniques of showing the Order
In their paper The integral of Riemann Xi Function, Lagarias and Montague show that the integral
$$\Xi_{\lambda}^{-1}(z)=2\int_0^{\infty}e^{\lambda u^2}\phi(u)\Big( \frac{\sin zu}{u}\Big)\;du$$
is ...
1
vote
1
answer
115
views
Zeroes of entire function on $\mathbb C^n$
Let $n\ge 2$ be an integer and let $f$ be an entire function on $\mathbb C^n$. Let $A$ be a subset of $\mathbb R^n$ with positive $n$-dimensional Lebesgue measure. Then if $f$ vanishes at $A$, this ...
5
votes
1
answer
305
views
Conjectured closed form of $\int_0^1 \frac{\text{Li}_2\left(\frac{x}{4}\right)}{4-x}\,\log\left(\frac{1+\sqrt{1-x}}{1-\sqrt{1-x}}\right)\,dx$
After reading some meta posts, I've decided to post this question on MathOverflow since I didn't receive any comments or answers on MSE
Certainly, I apologize for any oversight. Here's a more refined ...
6
votes
1
answer
208
views
Mellin-Barnes integral representation of the exponential function with a non-real argument
I have been studying a definite integral that I found out to be a particular (and possibly one of the simplest) case(s) of the arcane Mellin-Barnes integral. Solving this problem would lead to a ...
5
votes
0
answers
185
views
Reference request: Automorphisms of $\mathbb C\{x,y\}$ which preserve the equation of the cusp, $x^3 - y^2$
In my research I encountered automorphisms of the ring of convergent power series
$$\varphi: \mathbb C\{x,y\} \to \mathbb C\{x,y\},$$
which preserve $f = x^3 - y^2$, i.e. $\varphi(f) = f$. I'm ...
1
vote
2
answers
133
views
Problem in understanding maximum principle for subharmonic functions
I am reading subharmonic functions and their properties from the book From Holomorphic Functions to Complex Manifolds by Grauert and Fritzsche. Let me first define what a subharmonic function is.
...
5
votes
1
answer
136
views
Upper bound for the $n$-th derivative of a rational function $\frac{f}{f+g}$
Let $f$ and $g$ be real polynomials with nonnegative coefficients. Let
$$
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
135
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
92
views
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, ...
0
votes
1
answer
74
views
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
vote
0
answers
185
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
306
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
362
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?
$$\lim_{r\rightarrow1^{-}}\sum_{j=1}^{\infty}\...
0
votes
1
answer
122
views
Small phase approximation
Does anyone known how to prove that if $|\phi_k (r)| \ll 1$ for all $r$ and all $k=1,...,n\,$, the following equation
$$ 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
98
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
1
answer
133
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
162
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
134
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
159
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
300
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
473
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
136
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
73
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
636
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
111
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
vote
0
answers
55
views
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
299
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
130
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
111
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
81
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
80
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
$$|f(z)|\sim |z|^{-a},\qquad |z|\to \...
0
votes
0
answers
104
views
Affine scheme over ring of meromorphic functions with finite poles on unit circle
I am looking into the set $S$ of meromorphic functions with a finite number of poles on the unit circle (i.e., rational functions with poles on the unit circle). I assume that any $h\in S$ has the ...
3
votes
0
answers
80
views
Transformation of Julia set sequence emerging from meromorphic function
I consider a sequence of meromorphic functions on the Riemann sphere $f_k:\hat{\mathbb{C}} \to \hat{\mathbb{C}}$ for $k\in\mathbb{N}$ of the form
$$f_k(z)=\sum_{j=1}^{n_k}\dfrac{1}{(z-p_j)^{c_j}}$$
...
0
votes
0
answers
63
views
Estimating a multiple integral of complex-valued function
I am trying to find an estimate of the following sum:
$$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(...