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

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118
votes
46answers
39k views

Ways to prove the fundamental theorem of algebra

This seems to be a favorite question everywhere, including Princeton quals. How many ways are there? Please give a new way in each answer, and if possible give reference. I start by giving two: ...
0
votes
0answers
14 views

Zeroes of trigonometric-like function

Consider a function $f(z)=\cos(z)\cosh(az)+\sin(z)\sinh(bz)$ for $z\in \mathbb{Z}, a,b \in \mathbb{R}$. Denote D being a set {a,b} of such pairs of parameters that NOT ALL zeroes of corresponding ...
6
votes
3answers
175 views

Logarithms of matrices in the disk-algebra

It is easy to see that within the disk algebra $A(D)$ $$\Delta(z):= \begin{pmatrix} 1&0\\z&1 \end{pmatrix}\; \begin{pmatrix} 1&1\\0&1 \end{pmatrix}= \begin{pmatrix} ...
6
votes
1answer
221 views

Rotation invariance of an integral

Consider the integral depending on 2 parameters $$f(\tau,x):=\int_{-\infty}^{+\infty}\frac{dp}{\sqrt{p^2+1}}e^{-\sqrt{p^2+1}\tau+ipx},$$ where $\tau >0,x\in \mathbb{R}$. This integral absolutely ...
2
votes
0answers
51 views

Functions of form $f(z)/f(z^*)$

I am doing my research in mathematical physics, and in the process I am getting functions of complex variable $z$ of form $F(z) = \frac{f(z)}{f(z^*)}$ In my case $f$ doesn't have any interesting ...
18
votes
3answers
397 views

Does $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converge to zero when $\alpha$ is irrational?

I came across a problem concerning about the convergence of products. I wonder if the complex series $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converges to zero when $\alpha$ is irrational. Of course, ...
5
votes
0answers
84 views

Complex Stone-Weierstrass Type problem

I have come across this problem which resembles complex Stone-Weierstrass theorem except for a problem that the conjugate of the functions are not necessarily in the sub algebra. Suppose $\Omega$ is ...
2
votes
1answer
144 views

The spectrum and the tangent space of the algebra of holomorphic functions on a Stein manifold

Let $A$ be a Fréchet algebra over ${\mathbb C}$, and let us call the spectrum ${\tt Spec}[A]$ of $A$ the set of all characters, i.e. continuous multiplicative linear functionals $s:A\to{\mathbb C}$, ...
25
votes
3answers
2k views

What is special about polylogarithms that leads to so many interesting identities and applications?

I have heard that Polylogarithms are very interesting things. The wikipedia page shows a lot of interesting identities. These functions are indeed supposed to have caught the attention of Ramanujan. ...
8
votes
3answers
493 views

Can the topological algebra of analytic functions be endowed with a norm that defines the natural topology?

Right, so in my research in complex analysis I was puzzled by this question which may have a simple approachable answer that eludes me, but I am truly itching to find out and in need of it so I am ...
1
vote
0answers
34 views

explicit conformal map of sinus-shaped region

I wonder weather an explicit conformal map of a sinus-shaped region given by $\Re(z) \in [0, 1+\delta \sin(\Im(z)) ]$ onto say a strip or a ball is known (of course $0<\delta < 1$). Thank you.
0
votes
0answers
103 views

Contour integral of non holomorphic but continuous functions [closed]

Let us consider two functions $f(z)$ and $g(z)$, both holomorphic on a domain $U$ (a simply connected subset of $\mathbb{C}$). Thus, because of Cauchy's integral theorem, along any closed rectifiable ...
2
votes
1answer
86 views

On some curves of real values of a rational function

For given parameters $a_{1},\dots,a_{k}\in\mathbb{R}$, define the rational function $\phi:\mathbb{C}\to\mathbb{C}$ as $$\phi(z)=\frac{1}{z}-a_{1}z-a_{2}z^{2}-\dots-a_{k}z^{k}.$$ The domain of its real ...
2
votes
2answers
88 views

Analytic continuation of a specific integral with respect to a parameter

The following integral absolutely converges for $Re(z)<0$ and is analytic in this domain: $$F(z):=\int_{0}^{+\infty}\frac{\sin r}{r}e^{\sqrt{r^2+m^2}z}dr ,$$ where $m>0$ is fixed. Question. To ...
9
votes
4answers
931 views

What's the use of Malgrange preparation theorem?

The Malgrange preparation theorem,which is the $C^{\infty}$ version of the classical Weierstrass preparation theorem,says that if $f(t,x)$ is a $C^{\infty}$ function of $(t,x)\in\mathbb{R}^{n+1}$ near ...
2
votes
1answer
2k views

Relation between complex analysis and harmonic function theory [closed]

There are some theorems in harmonic function theory that resemble results in complex analysis, like: Holomorphic functions and complex functions are analytic; Cauchy's integral formula in complex ...
4
votes
1answer
259 views

Two similar integrals

Let $n$ be a given even positive integer. We have the following integral \begin{align} ...
7
votes
1answer
290 views

Equivalence of Branched Coverings

For equivalence of unbranched coverings of topological spaces, there is a criteria: Two coverings (unbranched) $p_1\colon Y_1\rightarrow X$ and $p_2\colon Y_2\rightarrow X$ are equivalent iff for ...
1
vote
0answers
74 views

Poincaré inequality for holomorphic line bundles

Let $M$ be a Riemann surface of genus >1, $g$ be an Hermitian metric on $M$. Let $E$ is a holomorphic negative line bundle over $M$, for example, the holomorhic tangent bundle of $M$. Let $h$ be an ...
2
votes
0answers
37 views

Estimates for hyperbolic metrics on nested surfaces

Suppose that $X$ and $Y$ are connected hyperbolic Riemann surfaces, with $X\subsetneq Y$. Let $\rho_{X,Y}$ be the density of the hyperbolic metric of $X$ with respect to that of $Y$; then $\rho_{X,Y} ...
21
votes
4answers
699 views

When are some products of gamma functions algebraic numbers?

I want to know when certain expressions of the form $ {\Gamma(r_1/m) \Gamma(r_2/m) \ldots \Gamma(r_j/m) \over \Gamma(s_1/m) \Gamma(s_2/m) \ldots \Gamma(s_j/m)} $ are algebraic numbers. These ...
2
votes
1answer
129 views

Blaschke Condition for hyperbolic lattices

For $r$, $s$, small positive integers, do the complex numbers on the unit disc (without the hyperbolic metric) corresponding to the vertices of the hyperbolic tiling with Schläfli symbol $\{r,s\}$ ...
4
votes
1answer
161 views

Artin approximation vs implicit function theorem in the class of analytic functions

I asked this on math stackexchange but I had no luck, so I am posting my question also here. I am not an algebraist so my question might be stupid. I am doing mainly complex analysis and recently I ...
1
vote
0answers
49 views

Complex integration over 1-singular chain

Let $f$ be a continuous function on $\mathbb{C}$ and assume that $\lim_{z\to \infty} zf(z) = \lambda.$ Let us note for all natural $n$ $$C_n = \{z \in \mathbb{C} : |z|=n\}.$$ Then, a usual fact of ...
1
vote
1answer
53 views

Can I apply Lagrange inversion theorem? [closed]

I want to invert the equation $$\eta = g(x)\sqrt{1+g'(x)^2}$$ to get $x$ as a function of $\eta$. Assume $g(0)=0$, $g'(0)=0$ and $g'(x)>0$ for $x>0$ (Think $g(x) = x^p$ for $p\geq 2$ integer). ...
1
vote
1answer
149 views

find solution of complex number recurrence equation

I have the following recurrence equation: $$(\mu\ n + \nu) f_{n} + J\Phi^{*} \sqrt{n+1}f_{n+1} + J\Phi\ \sqrt{n}f_{n-1} = 0$$ for complex numbers $f_{n}$ where $n = 0,1,2,3,...,\infty$ and complex ...
1
vote
1answer
146 views

A Generalization of growth exponents

The growth exponent of a function $f(\sigma + it)$ is defined as the least nonnegative real number $\psi(\sigma)$ satisfying $$ f(\sigma + it) \ll |t|^{\psi(\sigma) + \epsilon} $$ as $t \to \infty$, ...
0
votes
0answers
31 views

Normality criterion based on Brownian motion

Consider analytic family $\mathcal{F}$ btw domains $U,V\subset \mathbb{C}$. For any $f\in \mathcal{F}$ we have time-changed Brownian motion $f(B_{t})=\widetilde{B}_{\int_{0}^{t}|f(B_{s})|^{2}ds}$. So ...
0
votes
0answers
41 views

How to get Wiener-Hopf decomposition of this matrix function

I met a matrix function $M(z)$ that I need to get its Wiener-Hopf decomposition that $M(z) = M_+(z) M_-(z)$ with $M_+(z)$ and $M_-(z)$ being analytic inside and outside the unit circle respectively. ...
2
votes
1answer
129 views

Proof of Euler's reflection formula via rapidly decreasing Fourier series

Story I want to prove Euler's reflection formula by showing that \begin{equation*} f(s) = \sin(\pi s) \Gamma(s) \Gamma(1 - s) \end{equation*} is constant, where $s = \sigma + it$. It's easy to see ...
15
votes
2answers
2k views

How did Riemann calculate the first few non-trivial zeros of the zeta-function?

Does anyone know how Riemann calculated the first few non-trivial zeros of the Zeta function? I am wondering if he approximated the integral, $\frac{1}{2 \pi i} \int_{R} \frac{{\xi}^\prime(z)}{\xi ...
6
votes
1answer
303 views

Dominated convergence to characteristic function

Let $\phi_m(x):=\chi_{[0,1]} * \chi_{[0,1]} *...* \chi_{[0,1]}$ be the m -times convolution (so $m+1$ characteristic functions are involved). Then the Fourier transform of this function is given by ...
-1
votes
0answers
34 views

How to prove that arc segment vanishes

I have this integral: $$iNx^a\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{x^{Ne^{i\theta}}e^{i\theta}}{\Gamma(Ne^{i\theta}+a-m)\zeta(2Ne^{i\theta}+2a+n)\sin (\pi \left(a+Ne^{i\theta}\right))} ...
1
vote
2answers
77 views

Nonlinear PDE for a 2D foliation

I am trying to solve a 4th order nonlinear PDE for a real function $u(x,y)$ of two variables. It is too complicated to reproduce here but it exhibits the following two very nice properties: 1) if ...
1
vote
0answers
40 views

Modulus of Continuity for an Analytic Function on an Ellipse

This is a question which I stumbled upon while working on Legendre Polynomials, but it is actually a question in complex analysis. Consider: Given $f\in C^{\infty} (E)$, where $E_{\rho} \subseteq ...
11
votes
2answers
375 views

regular polygon question

Let $a_1,a_2,\ldots,a_n$ be distinct points on the complex plane $\mathbb{C}$ and $L$ be a circle in $\mathbb{C}$ such that $$f(z):=\sum_{i=1}^n|z-a_i|^{2n-2}$$ is constant on $L.$ Could somebody ...
7
votes
0answers
159 views

Biholomorphic neighborhoods of the boundary of Stein domains

Let $(X_1,J_1)$ and $(X_2,J_2)$ be Stein domains with the same contact boundary $(Y,\xi)$. Under what conditions does there exist a biholomorphism between a neighborhood of their respective boundaries ...
4
votes
1answer
160 views

Embed a bordered Riemann surface into punctured Riemann surfaces?

Let $U$ be a bordered Riemann surface of genus $g$ with $n -1$ punctures and one hole (i.e., the border has one connected component). For any punctured Riemann surface $\Sigma$ of genus $g$ with $n$ ...
3
votes
1answer
83 views

Wiener-Hopf factorization of matrices

Given a $2\times2$ matrix, which entries are functions in the complex plane $$\hat{A}(z)=\left(\begin{array}{cc}a(z)&b(z)\\c(z)&d(z)\end{array}\right)$$Where $a(z),b(z),c(z)$ and $d(z)$ are ...
2
votes
0answers
108 views

The boundedness of an entire function along the imaginary axis

I am looking for an answer in the affirmative or the negative concerning the asymptotics of an entire function. The question, relatively simple to state, has proven very taxing to solve and requires ...
3
votes
1answer
89 views

$L^p$ norm of boundary values of holomorphic function

I am looking for an estimate of the following form: Suppose that $D\subset \mathbb{C}$ is a simply connected domain. Suppose that $F$ is holomorphic and bounded on $D$ and can be holomorphically ...
1
vote
0answers
62 views

Explicit formula of biholomorphism between the rectangle and unit disk [closed]

From the Riemann mapping theorem we know that there exists a biholomorphism between the rectangle $R$ and the unit disk $D$, can we write this biholomorphic map explicitly?
0
votes
1answer
100 views

How to calculate the expected value of complex-valued random variable? [closed]

Suppose $\theta_1,\theta_2,\cdots, \theta_n$ are independent and identically distributed (i.i.d.) real-valued random variables and here we specifically consider the uniform distribution in the ...
3
votes
0answers
98 views

Siegel's article “The volume of the fundamental domain for some infinite groups”: trouble with understanding computations

This is the article I mentioned. While the idea of what Siegel is doing in order to compute the volume of the fundamental domain described in the article (the very first one, for there are discussed ...
6
votes
1answer
112 views

Convex Hull of univalent functions and Bieberbach Conjecture

Consider the class $S$ of univalent functions on the unit disk $D$ normalized so that $f(0)=0$ and $f'(0)=1$. Each function in $S$ satisfy the Bieberbach conjecture, that is the $n$-th coefficient in ...
3
votes
0answers
83 views

State of the art for univariate complex polynomials factorization with algebraic coefficients

Let $\mathbb{K}:=\overline{\mathbb{Q}}$ be the field of algebraic numbers. We choose to represent an element of $\mathbb{K}$ as its minimal monic polynomial, which is a vector in some $\mathbb{Q}^n$. ...
-4
votes
1answer
177 views

Proof of formula for $\pi$ in terms of infinite number [closed]

What is the shortest proof of the formula $\pi=\frac{\infty\sqrt[\infty]{-1}-\infty}{\sqrt{-1}}$? Here we choose the principal branch of the root. Hopefully a fairly elementary proof can be provided ...
5
votes
2answers
489 views

Holomorphy of a function with values in a Hilbert space

I asked the same question on MathStackExchange. EDIT: This question has now a open bounty worth +50 reputation on MSE. Denote by $\mathbb C^\infty $ the Hilbert space $\ell^2 (\mathbb C)$. Fix $1\leq ...
6
votes
2answers
158 views

Criterion for deciding the conformal class of a metric on a complete surface

For orientable closed Riemannian surfaces $(S,g)$, there is a constant curvature metric $\overline{g}$ on $S$ that is conformal to $g$ in the sense that $\overline{g} = e^ug$ for some smooth function ...
11
votes
3answers
515 views

How many isolated roots can a polynomial in $z$ and $\overline{z}$ have?

This is probably well-known by I can't find it now online. My guess is that if the degree is $n$ then it's $2n$ but it's just a hunch. EDIT: This is an edited version. Before I asked about roots ...