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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 $...
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 $g$ ...
158 votes
8 answers
7k views

Resources for mathematics advising.

This question is possibly ill-advised. (If it is not right for this site I will delete it.) I, suddenly, have students. It is very clear to me that there is nothing in my education that has ...
58 votes
4 answers
5k views

Advice for PhD Supervisors

My first PhD student is having his viva tomorrow. Hence, I began contemplating a bit about the whole process of supervising. One thing I realized is that while there seems to be plenty of advice for ...
74 votes
51 answers
28k views

An example of a beautiful proof that would be accessible at the high school level?

The background of my question comes from an observation that what we teach in schools does not always reflect what we practice. Beauty is part of what drives mathematicians, but we rarely talk about ...
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 ...
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(...
3 votes
1 answer
156 views

Green potential and Hölder continuity

Assume that $U$ is the unit disk and $g\in L^{3/2}(U)$. Define $$f(z) = \int_{U} \log\left|\frac{z-w}{1-z\bar w}\right|g(w)\frac{du \, dv}{\pi}, \ \ w=u+iv.$$ Is there an elementary proof of the fact ...
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 ...
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 ...
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{...
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 $\...
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\...
5 votes
0 answers
322 views

Approximating $\zeta^{(r)}(s)$ by a sum

Let $\eta:[0,\infty)\to [0,\infty)$ be compactly supported, continuous and piecewise $C^1$, with its derivative $\eta'$ being of bounded variation. It is completely unsurprising that one can prove (...
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 ...
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 ...
3 votes
0 answers
102 views

Stein manifolds admitting uniform strictly plurisubharmonic exhaustion functions

Let $(X,\omega)$ be a Kähler manifold. Call $X$ uniformly Stein with respect to $\omega$ if there exists an exhaustion map $\phi:X\to \mathbb{R}$ such that $i\partial\bar\partial \phi \ge \epsilon\...
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 $...
-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: \...
4 votes
3 answers
714 views

Jordan curve theorem for cylinders

Hello, I would like to know if the following result is true: Let $A,B$ be two embedded circles in $S^2$ which do not intersect and let $C$ be the $\textit{closed}$ region bounded by $A$ and $B$ (...
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 ...
52 votes
3 answers
6k views

Is the Riemann zeta function surjective?

Is the Riemann zeta function surjective or does it miss one value?
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 ...
7 votes
6 answers
1k views

Another chicken or egg: sequence or series

This is a side question which is more motivated by teaching than research. First, I am trying to convince myself that sequences appear before series (as numerical approximations to "interesting" ...
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 $\epsilon, \mu >0$, ...
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 ...
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 votes
2 answers
561 views

$\frac{\partial f}{\partial \overline{z}}=0$ in distributional sense implies $f$ is holomorphic

Let $f=(u,v)\in \mathscr{D}'(U,\mathbb{C})$ be a distribution, where $U\subset\mathbb{C}=\mathbb{R}^2$ is an open set and $u$ and $v$ are the projection of $f$ onto the real and imaginary axis (ie $\...
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 ...
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 ...
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 votes
2 answers
316 views

Is a local isometry of the hyperbolic plane the restriction of a global isometry?

The origin question: Let $\Omega \subset \mathbb{H}^2$ be a domain of the hyperbolic plane $\mathbb{H}^2$. Let $u: \Omega \to \mathbb{H}^2$ be injective and an isometry from $\Omega$ to its image. ...
0 votes
1 answer
138 views

Is $\Gamma(z,1)\not=0$ for all $z$ with $\Re(z)<0$?

I found this paper online which appears to present zeros of the incomplete gamma function within the right half plane. It makes me think that there are no zeros in the left half plane. Not sure how to ...
7 votes
1 answer
269 views

Efficiently computing $\sum_k x^{k^2}$ modulo $p$

Let $p$ be prime. There is a whole host of "large" degree polynomials that can be computed efficiently modulo $p$. I was wondering if: $$q(x) = \sum_{k=0}^{p-1} x^{k^2}$$ is a polynomial ...
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 ...
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 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 ...
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 ...
0 votes
1 answer
1k views

1895 Math Trip problem on primitive roots of unity

How to prove that if $\theta _1,\theta _2,\theta _3$ be the arguments of the primitive roots of unity, $\sum \cos p\theta = 0$ when $p$ is a positive integer less than $\dfrac {n} {abc\ldots k}$, ...
74 votes
15 answers
18k views

$f(f(x))=\exp(x)-1$ and other functions "just in the middle" between linear and exponential

The question is about the function $f(x)$ so that $f(f(x))=\exp (x)-1$. The question is open ended and it was discussed quite recently in the comment thread in Aaronson's blog here http://...
6 votes
2 answers
348 views

Does there exist a framework for determining if a power series is "differentially algebraic"

It is a well studied problem to take a function $f$ expressed (usually expressed as a solution to a differential equation w/ some initial conditions) and ask if it has an "elementary closed form&...
0 votes
0 answers
108 views

A surprising result with the Riccati difference equation

I was looking at the Riccati difference equation with positive and negative indices $$ R_n=\frac{aR_{n-1}+b}{cR_{n-1}+d}\quad n\in[0,N]\\ R_n=\frac{-dR_{n+1}+b}{cR_{n+1}-a}\quad n\in[-N,0]\\ $$ along ...
2 votes
0 answers
110 views

Quotient of integral representation of archimedean exterior square L-function

Let $\psi$ denote a non-trivial additive character of $\mathbb{R}$ and $r=2n$ be a positive even integer. Let $(\pi,V)$ denote an irreducible generic admissible Casselmann-Wallach representation of $...
1 vote
0 answers
125 views

Interpolating sequences are strongly separated

I am reading Agler and McCarthy's Pick Interpolation and Hilbert Function spaces. In Chapter 9, titled "Interpolating Sequences", the authors say that every interpolating sequence is ...
0 votes
1 answer
117 views

Sufficient conditions for ensuring that a monic polynomial in $\mathbf{Z}[x]$ possesses exclusively simple roots

I am seeking sufficient conditions to ensure that a monic polynomial, denoted as $f$ in $\mathbf{Z}[x]$, possesses exclusively simple roots. Based on an old paper (this reference), it has been ...
2 votes
1 answer
335 views

Combinatorial meaning of a binomial expansion

Let $F$ be a generating function $F(x) = \sum_{i=0}^\infty f_i x^i$, and suppose that we can do operations formally without worrying about convergence issues. Define the coefficients \begin{gather*} ...
5 votes
1 answer
315 views

Modeling the interior and exterior of a polygonal region on the Riemann sphere using Schwarz-Christoffel mappings

I am thinking about the following. I have been involved in a research project involving static magnetic fields inside and outside a polygonal magnetic material. You ended up trying to find a couple of ...
6 votes
0 answers
165 views

Fourier transform and Hodge-$*$ operator

Suppose I have a full-rank lattice $\Lambda\subset\mathbf{C}$. Then the classical Poisson summation formula says $$\sum_{\lambda\in\Lambda}f(\lambda)=\sum_{\lambda\in\Lambda'}\widehat{f}(\lambda)$$ ...
9 votes
1 answer
661 views

Is anything known about the power series $\sum x^p$ for $p$ prime?

I'm interested in information about the power series $$\sum_{\text{$p$ prime}} x^p$$ and the related power series $$\sum_{n=1}^\infty (-1)^n x^{p(n)}$$ where $p(n)$ is the nth prime. Immediately, the ...

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