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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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 $...
Terry Tao's user avatar
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-2 votes
2 answers
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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(...
Bo Jonsson's user avatar
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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(...
Gabriel Palau's user avatar
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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 ...
Tardis's user avatar
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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 ...
Ali's user avatar
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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 ...
psubodiosa's user avatar
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 ...
Aobara's user avatar
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1 vote
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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 \...
mathim1881's user avatar
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 \...
xen's user avatar
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6 votes
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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{...
Tian Vlašić's user avatar
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 ...
Tokita Ohma's user avatar
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 ...
Bazin's user avatar
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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 ...
Martin.s's user avatar
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6 votes
1 answer
207 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 ...
ThomasJr's user avatar
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 ...
red_trumpet's user avatar
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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. ...
Anacardium's user avatar
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^{(...
xen's user avatar
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0 answers
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“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 ...
JustSomeGuy's user avatar
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, ...
psubodiosa's user avatar
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 $\...
Factorial_zero's user avatar
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 $...
Thomas Kurbach's user avatar
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 $$ 2\sum_{k=1}^{\infty}(-1)^k\,k^5\,J_{2k}(x) = \frac{x^2}{4}\left[x\,J_1(x)-J_0(x)\right] \label{1}\tag{1} $$...
Alessandro Pini's user avatar
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}\...
T. Amdeberhan's user avatar
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 ...
Tomaž Požar's user avatar
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 ...
GBA's user avatar
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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 ...
Factorial_zero's user avatar
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 ...
Thomas Kurbach's user avatar
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: $...
SnowRabbit's user avatar
-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 ...
ironmanaudi's user avatar
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(...
Claudio's user avatar
  • 11
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'...
S.H.W's user avatar
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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 ...
ironmanaudi's user avatar
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 $[...
mathematics is all's user avatar
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 ...
Vik78's user avatar
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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....
Tian Vlašić's user avatar
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 ...
Math1016's user avatar
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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 ...
pie's user avatar
  • 139
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 ...
Jens Fischer's user avatar
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 ...
Invariance's user avatar
-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 ...
free_lancer's user avatar
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 ...
Bo_Y's user avatar
  • 637
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 $d$ ...
Yee Neil's user avatar
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 ...
LYJ's user avatar
  • 11
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 \...
gm01's user avatar
  • 325
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, ...
ashK's user avatar
  • 83
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 $...
knuth's user avatar
  • 29
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 \...
Medo's user avatar
  • 676
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 ...
Jens Fischer's user avatar
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 $$f_k(z)=\sum_{j=1}^{n_k}\dfrac{1}{(z-p_j)^{c_j}}$$ ...
Jens Fischer's user avatar
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(...
Anish Ray's user avatar
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