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

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-2
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0answers
62 views

Prove a function is entire

Let $f$ function such that $\int_{R^m} (1+|y|)^N|f(y)|dy <\infty$. Consider a function $g(z) = \int h(y,z) f(y) dy,$ where $|h(y,z)| \leq C e^{|z||y|}$ with $z\in C^m$ and ...
5
votes
1answer
192 views

Is polynomial convexity a topological invariant?

Is the property of being polynomially convex a topological invariant? In other words, assume that $M$ and $N$ are two compact subset of $n$-dimensional complex Euclidean space which are homeomorphic. ...
1
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0answers
68 views

Moving from $\Re(s) = 1+\epsilon$ to $\Re(s) = \frac{1}{2}$ in the proof of the Weil-Guinand explicit formula

In all proofs of the Weil-Guinand explicit formula, there's this step (this is from Paul Garrett's notes): Now consider this: (1) $\frac{\zeta^\prime(s)}{\zeta(s)}$ has poles at $s=1$ and ...
0
votes
1answer
43 views

Absolutely continuous and rectifiable boundary

Assume that $\gamma$ is a Rectifiable curve in $\mathbf{C}$ and ssume that $f$ is a bounded holomorphic function on the unit disk $U$ such that if $z_n$ converges to a boundary point of $\mathbf{U}$, ...
2
votes
1answer
136 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 $\{r,s\}$ ...
1
vote
0answers
54 views

How quickly can we mutliply Cayley-Dickson hypercomplexes?

Assuming that all of the coordinates of two Cayley-Dickson Hypercomplex numbers are non-negative integers less than a prime $p$, how quickly can we multiply these numbers? I'm also interested in what ...
18
votes
2answers
906 views

How to prove that $e^{zw}$ can not be written as a rational expression in functions in $z$ and in $w$?

Let $K$ be the field of fractions of $\mathbb{C}[[z]]\otimes_{\mathbb{C}}\mathbb{C}[[w]]\subset \mathbb{C}[[z, w]].$ Given a formal power series in $t, f\in \mathbb{C}[[t]],$ is there any simple ...
4
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1answer
194 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 ...
3
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4answers
1k views

Classification compact Riemann Surfaces

I know that all compact Riemann surfaces with the same genus are topologically equivalent. Moreover they are diffeomorphic. But are they biholomorphic, too? In other words, is the complex structure ...
2
votes
1answer
224 views

upper bound on derivatives of a function defined on an arc

This is a simple question I asked in math.SE last month but unfortunately no one gives any comment. So I decided to try some luck here. You can skip examples below and read from "General setting" at ...
-4
votes
1answer
275 views

Proof of formula for $\pi$ [closed]

The number $\pi$ can be expressed as $\pi=\lim_{n\to\infty} \frac{n\sqrt[n]{-1}-n}{\sqrt{-1}}$ or more poetically $\pi=\frac{\infty\sqrt[\infty]{-1}-\infty}{\sqrt{-1}}$. Here we choose the principal ...
6
votes
1answer
317 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 ...
2
votes
2answers
149 views

Zeroes of trigonometric-like function

Consider a function $f(z)=\cos(z)\cosh(az)+\sin(z)\sinh(bz)$ for $z\in \mathbb{C}, a,b \in \mathbb{R}$. Denote $D\subseteq \mathbb{R}^2$ being the set of such pairs $(a,b)$ of parameters so that NOT ...
5
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0answers
148 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 ...
0
votes
2answers
208 views

Absolute value inequality with complex numbers

Following a problem I found on mathstack, with no solution, and no comment, so I think this inequality is not easy, so I post it here (because I think there are more some good math job, maybe someone ...
13
votes
3answers
455 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 ...
0
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0answers
72 views

Continuation to holomorphic function

Let G be a k-fold connected riemann surface with boundary given by $k >2$ non-intersecting Jordan-curves and $\alpha : \partial G \longrightarrow S^1$ a continuous map. Now i constructed a ...
2
votes
1answer
76 views

Octahedron and System of trigonometric equations

Could somebody help me to prove the following? $$\sum_{k=1}^6 \cos(2 \theta_k) (\cos(2\phi_k)-1))=0$$ $$\sum_{k=1}^6 \sin(2 \theta_k) (\cos(2\phi_k)-1))=0$$ $$\sum_{k=1}^6 \cos (\phi_k)=0$$ ...
1
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0answers
96 views

Applications of Iss'sa's theorem on homomorphisms between algebras of meromorphc functions

In Remmert's book Funktionentheorie II, the following theorem, apparently due to Hironaka under the pseudonym Iss'sa, is proved: Let $U,V \subseteq \mathbb{C}$ be open subsets. Every ...
0
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0answers
32 views
1
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2answers
195 views

How to evaluate the following integral

Would anyone please let me know how to compute the following integral: $$\int_{-\infty}^{+\infty}\frac{a\log(t^2+1)}{t^2 + a^2}dt,$$ here $a > 0$.
4
votes
3answers
2k views

Branched coverings of Riemann surfaces with specified branch points.

Today I showed, using some ad hoc algebraic topology, that if $\Sigma$ is a Riemann surface and $\mathfrak{p} \subset \Sigma$ is a finite set of points, then there is another Riemann surface $S$ and a ...
0
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1answer
78 views

Branches of the tetration function

Letting $\eta = e^{1/e}$ where $e$ is Euler's constant, there exists a function $F(z)=\, ^z \eta$ with the following relevant properties. (I won't bother showing the existence of this function, or the ...
1
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0answers
75 views

extension for a complex operator

Let be $\lambda>0$. Put $$ L_{\lambda}=\Big[-\frac{\partial^{2}}{\partial z \partial \overline{z}}+\lambda^{2}|z|^{2} +\lambda\Big(\overline{z}\frac{\partial}{ \partial ...
117
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: ...
6
votes
3answers
193 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} ...
7
votes
1answer
251 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
57 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
460 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, ...
2
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1answer
157 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
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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. ...
9
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3answers
506 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
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0answers
35 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.
2
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1answer
93 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
89 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 ...
10
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4answers
940 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
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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
263 views

Two similar integrals

Let $n$ be a given even positive integer. We have the following integral \begin{align} ...
7
votes
1answer
293 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
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0answers
77 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
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0answers
45 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
702 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 ...
1
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0answers
50 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
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1answer
54 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
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1answer
154 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
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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
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0answers
32 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
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0answers
42 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
135 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
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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 ...