All Questions
3,560 questions
3
votes
1
answer
459
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?
$$\lim_{r\rightarrow1^{-}}\sum_{j=1}^{\infty}\...
- 43.2k
0
votes
1
answer
128
views
Small phase approximation
Does anyone known how to prove that if $|\phi_k (r)| \ll 1$ for all $r$ and all $k=1,...,n\,$, the following equation
$$ S=\left|\int_0^\infty A(r)e^{-i[\phi_0(r)+\sum_{k=1}^n \phi_k(r)]} dr \right|^2 ...
- 13
1
vote
1
answer
106
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 ...
- 167
2
votes
1
answer
148
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
186
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 ...
- 513
2
votes
0
answers
150
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:
$...
- 263
-1
votes
1
answer
215
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 ...
- 149
1
vote
1
answer
110
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(...
- 11
0
votes
1
answer
419
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'...
- 61
4
votes
3
answers
685
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 ...
- 149
2
votes
0
answers
146
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
89
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 ...
- 658
7
votes
2
answers
720
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....
- 405
0
votes
0
answers
121
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 ...
- 369
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 ...
- 541
1
vote
1
answer
105
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 ...
- 91
-3
votes
2
answers
318
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 ...
- 317
1
vote
1
answer
152
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 ...
- 637
3
votes
1
answer
119
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 $d$ ...
- 97
0
votes
1
answer
65
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 \...
- 327
1
vote
0
answers
115
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, ...
- 151
1
vote
0
answers
86
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 $...
- 33
0
votes
0
answers
94
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
$$|f(z)|\sim |z|^{-a},\qquad |z|\to \...
- 852
0
votes
0
answers
109
views
Affine scheme over ring of meromorphic functions with finite poles on unit circle
I am looking into the set $S$ of meromorphic functions with a finite number of poles on the unit circle (i.e., rational functions with poles on the unit circle). I assume that any $h\in S$ has the ...
- 91
3
votes
0
answers
89
views
Transformation of Julia set sequence emerging from meromorphic function
I consider a sequence of meromorphic functions on the Riemann sphere $f_k:\hat{\mathbb{C}} \to \hat{\mathbb{C}}$ for $k\in\mathbb{N}$ of the form
$$f_k(z)=\sum_{j=1}^{n_k}\dfrac{1}{(z-p_j)^{c_j}}$$
...
- 91
0
votes
0
answers
66
views
Estimating a multiple integral of complex-valued function
I am trying to find an estimate of the following sum:
$$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(...
- 309
3
votes
2
answers
646
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 ...
- 43
7
votes
1
answer
488
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 ...
- 6,367
1
vote
0
answers
110
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 ...
- 513
2
votes
2
answers
268
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 $\...
- 605
2
votes
0
answers
88
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
152
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 ...
- 438
6
votes
0
answers
220
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 ...
- 370
0
votes
0
answers
74
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 $g$ ...
- 1,169
1
vote
0
answers
65
views
Complex integration in a separable EDO for an eigenvalue problem
I'm trying to prove a result from Leon Cohen's book "Time-frequency Analysis", Chapter 18. Namely, I want to verify the solution of the eigenvalue problem $$\mathcal{C}s(t) = c s(t)$$ for ...
2
votes
1
answer
108
views
Mandelbrot boundary and component of $\infty$
Let $M$ be the Mandelbrot set, and $\partial M$ its boundary. So $\partial M$ is the set of those points $z\in M$ such that every neighborhood of $z$ contains a point of $\mathbb R^2\setminus M$.
Let $...
- 3,317
-1
votes
2
answers
87
views
Limits of integral series
Suppose we have the series of functions:
\begin{equation}
F(x)=\sum_{n=1}^{\infty} f_n(x)
\end{equation}
where convergence is uniform.
Additionally, consider the partial functions of the series:
\...
1
vote
0
answers
82
views
Finiteness of theta vanishing in the KP direction for locally planar curves
I believe the main question is Question 2 at the end, and for experts it might be completely okay to skip directly to it (assuming I'm not saying any nonsense).
My motivation comes from pure algebraic ...
- 318
1
vote
1
answer
272
views
Let $u$ be harmonic on domain $D\subset \mathbb R^d$, how far can we extend $u$ holomorphically?
If $u$ is harmonic then it is real analytic so then it can be extended locally holomorphically. I also know that if $u$ is harmonic on a ball in $\mathbb R^d$ we have that the radius of convergence is ...
- 1,914
0
votes
0
answers
66
views
Meromorphic functions converging in measure
Let $f_1, f_2, \ldots$, and $g$ be measurable complex-valued functions on the open unit disk. We say that the sequence $f_1, f_2, \ldots$ converges in measure to $g$ if, for all $\epsilon, \mu >0$, ...
- 974
0
votes
0
answers
107
views
Evaluating a matrix Pick function via its integral representation
In the proof of Theorem 3.1 of the paper Inequalities for M-matrices, Ando evaluates a matrix function (see equation boxed in orange below) via an integral representation of a Pick function (see ...
- 1,719
4
votes
0
answers
112
views
Elliptic integral as quantity associated with Riemann surface?
There are many elliptic integrals, so to show my point let me
just pick one of them (complete elliptic integral of the first
kind [1]):
$$K(k) = \int_{0}^{1} \frac {dx} {\sqrt{(1-x^{2})(1-k^{2}x^{2})}}...
- 5,230
4
votes
2
answers
374
views
Abel–Plana formula with fractional offset
The Abel–Plana formula compares the sum $\sum_{n=0}^\infty f(n)$ to the integral $\int_0^\infty f(x)\,dx$,
\begin{equation}
\sum_{n=0}^{\infty}f\left(n\right)-\int_{0}^{\infty}f\left(x\right)dx=\frac{...
- 188k
5
votes
1
answer
180
views
Is there a generalisation of the Vivanti-Pringsheim theorem for several variables?
The Vivanti-Pringsheim theorem states that if $f(z)$ has a power series with non-negative coefficients and a radius of convergence $R > 0$, then it has a singularity at $R$. So to find the radius ...
- 887
3
votes
0
answers
111
views
What is known about the analytic continuation of Maz'ya's modified harmonic zeta function $\sum_{n=1}^{\infty} e^{-zH_n}$?
Question:
If we let $H_n = \sum_{k=1}^{n} \frac{1}{k}$ be the harmonic numbers then we can consider the modified zeta function
$$ f(z) = \sum_{n=1}^{\infty} e^{-zH_n } = \sum_{n=1}^{\infty} e^{-z(\ln(...
- 2,689
11
votes
3
answers
1k
views
"Simple" integral equation
Let $H(z)$ be a continuous solution of the problem
$$
H(z)=\frac1{1-z}\int_z^1 \frac{2\zeta}{1+\zeta} H(\zeta^2)\,d\zeta,\ \ \ z\in[0,1);\ \ \ H(1)=1.
$$
Is it true that $H(0)=1-\ln2$? The question ...
- 283
3
votes
1
answer
174
views
$n$-th root of meromorphic functions of several complex variables
Let $f:\Omega\rightarrow\mathbb{C}$ be a "nice" meromorphic function of several complex variables on some domain $\Omega$. I wonder if the following claim is true.
Claim. $f$ admits a global ...
- 1,024
1
vote
1
answer
119
views
weakly separated sequences in RKHS are separated by Gleason metric
I am reading Agler and McCarthy's Pick Interpolation and Hilbert Function spaces. In Chapter 9, the authors ask to observe that weakly separated in a Reproducing kernel hilbert space implies separated ...
- 151
12
votes
4
answers
929
views
Interesting examples of systems of linear differential equations with constant coefficients
In this paper, Gian-Carlo Rota wrote:
A lot of interesting systems with constant coefficients have been discovered in the last thirty years: in control, in economics, in signal
processing, even in ...
2
votes
1
answer
271
views
Irreducibility of an explicit complex projective variety
Let $Y\subset \mathbb P^n_\mathbb C$ be a subvariety defined by a series of homogeneous polynomials $f_1, \ldots, f_t$. Is there an effective way to determine the irreducibility of $Y$ as an algebraic ...
- 279