**7**

votes

**2**answers

308 views

### Periodicity in iterated powers of sin, cos, exp

Given a complex number $z$, consider the sequence
\begin{align*}
a_0 & = 1\\
a_1 & = (cos(1))^z\\
a_n & = (cos(a_{n-1}))^z
\end{align*}
This question is about trying to understand ...

**2**

votes

**1**answer

154 views

### Division and multiplication that preserve Euclidean norms

I am looking for ways to define
$$\frac{1}{x}\in \mathbb{R}^n\quad \quad and\quad \quad x\cdot y\in \mathbb{R}^n ,$$
where $x,y\in \mathbb{R}^n$ such that
...

**13**

votes

**2**answers

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 ...

**1**

vote

**2**answers

196 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 $a > 0$.

**2**

votes

**1**answer

137 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 $\{r,s\}$ ...

**14**

votes

**1**answer

612 views

### Is there a Serre intersection formula in analytic geometry?

There is the famous Serre intersection formula in algebraic geometry using the Tor functor (see for example here). I would like to know if there is such a formula in analytic (i.e. complex) geometry. ...

**7**

votes

**2**answers

986 views

### Does module Hom commute with tensor product in the second variable?

Let $A$ be a commutative ring, and $L, M, N$ be $A$-modules. Then is it true that
$$\text{Hom}_A (L, M)\otimes_A N \cong \text{Hom}_A (L, M\otimes_A N)$$
as $A$-modules?
(Note that there is a ...

**1**

vote

**2**answers

185 views

### Complex structure on a punctured torus giving a complex structure on the torus?

Can anyone provide an idea of the proof or a reference of the fact that a complex structure on the once punctured torus extends to one on the torus?
In other words, the Teichmuller space of the ...

**1**

vote

**1**answer

87 views

### What is the Fano index for Hermitian symmetric spaces of compact type?

As we know Hermitian Symmetric spaces of compact type are all Fano picard number one, we can talk about his Fano index. Suppose $X$ is one of those Hermitian symmetric spaces, $L$ is the generator of ...

**2**

votes

**1**answer

167 views

### Is the set of entire functions Borel in the space of analytic functions?

$\def\bbR{\mathbb R}\def\bbC{\mathbb C}\def\scrT{\mathscr T}\def\ssp{\kern.4mm}
$More specifically, I ask whether $S$ be a Borel set in the topological space $(\Omega,\scrT)$ in the following ...

**26**

votes

**1**answer

787 views

### Stone-Weierstrass theorem for holomorphic functions?

The Stone-Weierstrass theorem has an analog for the algebras of smooth functions, called
Naсhbin's theorem: An involutive subalgebra $A$ in the algebra ${\mathcal C}^\infty(M)$ of smooth ...

**5**

votes

**0**answers

1k views

### Asymptotic Robin inequality and RH [closed]

There exists several equivalent formulations of RH. Among them, there is a criterion of Robin that describes a bound on the growth rate of the sum-of-divisors function $\sigma$.
Apparently yesterday ...

**9**

votes

**2**answers

549 views

### Fourier transform of the critical line of zeta?

This was asked on MSE and got a lot of upvotes but no answers, so I'm posting it here.
Is there a known expression for the (distributional) Fourier transform of the Riemann zeta function, taken along ...

**3**

votes

**1**answer

73 views

### Assuming admissible functions $\rho$ are continuous in definition of conformal modulus

It's stated in Väisälä's 'Lectures on n-dimensional quasiconformal mappings' (p. 20) that, in the geometric definition of a quasiconformal mapping, that the modulus of a family of curves associated to ...

**1**

vote

**0**answers

49 views

### Starlike curve tangent condition

Assume that $\gamma$ is a starlike Jordan curve in the complex plane w.r.t. 0. Let $\alpha\in (0,\pi/2]$. For each $z\in\gamma$, $z\neq x$, we let $\alpha(z,x)$
denote the acute angle which the ...

**1**

vote

**1**answer

74 views

### Inverse error function in Hardy space?

Let $\mathrm{erf}(x) := \frac{2}{\sqrt{\pi}} \int_{-\infty}^x \exp(-t^2) \, dt$ be the error function $\mathrm{erf}: \mathbb{R} \to (-1,1)$. It is monotonously increasing and therefore has an inverse ...

**2**

votes

**0**answers

119 views

### Integral representation of the complex homogeneous polynomial $z_1\cdots z_n$

Consider the transform (see e.g., (5.1) in this paper):
\begin{equation*}
\Lambda_\mu(q)(z) := \int_{\Delta_n} ...

**1**

vote

**0**answers

121 views

### Homeomorphism of fibers of holomorphic maps

EDIT (after the comment by Jason Starr): Let $X$ be a complex algebraic (or, more generally, analytic) variety, possibly singular and non-compact. Let $f\colon X\to D^*$ be a proper algebraic morphism ...

**1**

vote

**1**answer

180 views

### Jensen formula in $\mathbb{C}^n$?

Let $f:\mathbb{C}\to\mathbb{C}$ be an entire function with zero set $X\subset \mathbb{C}$. Jensen's formula reads
$$
\log(|f(0)|)+\int_0^R\frac{|X\cap B_t(0)|}{t}dt = ...

**0**

votes

**1**answer

79 views

### On a theorem by Mooney and Khavin on the weak sequential completeness of the predual of $H^\infty(\mathbb{D})$

There is a theorem by Mooney http://msp.org/pjm/1972/43-2/pjm-v43-n2-p.pdf#page=185 and independently proved by Havin which says that the predual of $H^{\infty}(\mathbb{D}),$
$L^{1}/H^{1}_{0}$ is ...

**12**

votes

**2**answers

565 views

### No nonconstant coprime polynomials $a(t)$, $b(t)$, $c(t) \in \mathbb{C}[t]$ where $a(t)^3 + b(t)^3 = c(t)^3$

See David Speyer's answer here.
I saw Brian Conrad give an excellent one hour talk to undergraduates where he proved that there do not exist nonconstant, relatively prime, polynomials $a(t)$, ...

**27**

votes

**3**answers

811 views

### A point set of power series with coefficients in {-1, 1}. Connected or not?

Let $z$ be a fixed complex number with $|z|<1$ and consider the set
$$X_z := \Big\{\sum\limits_{i=1}^{\infty} a_i z^i \ \Big|\ a_i\in \{-1,1\} \forall i\Big\}.$$
What can be said about the set $M$ ...

**4**

votes

**0**answers

108 views

### classification of homogenous complex manifolds

Suppose $X$ is a complex manifold (doesn't assume it's Kahler), and it's holomorhpic automorphism group is transitive. My question is that is there any classification of those manifolds ?

**2**

votes

**0**answers

54 views

### Hadamard Product of specific type of power series

I am consider the power series of the form $$F_n(t):=\frac{1}{\prod_{i=1}^n(1-t^i)}.$$ Given two power serires $A(t)=\sum_{i\ge 0}{a_it^i}$ and $B(t)=\sum_{i\ge0}{b_it^i}$, their Hadamard product is ...

**13**

votes

**3**answers

429 views

### Completeness of nonharmonic Fourier Series

I have the following question:
The Exponential System $(\exp(2\pi i n \cdot ))_{n\in \mathbb{Z}}$ constitutes an orthonormal basis of $L^2([-1/2,1/2])$.
Thus, certainly the oversampled system ...

**0**

votes

**2**answers

139 views

### Explicit analytic function with modulus asymptotic to $\Re z+\Im z$

Is there a simple and explicit continuous function $f\colon[0,\infty)^2\to\mathbb C$ such that $f$ is analytic on $(0,\infty)^2$ and $|f(x+iy)|/(x+y)\to1$ as $x+y\to\infty$, where ...

**11**

votes

**1**answer

434 views

### $\pi e$ and an unfamiliar polynomial

Ever since my exposure to this integral involving $\pi e$, I've conjectured and set about evaluating the possible nature of the following integral
$$\int_0^1 x^m \sin(\pi x) x^x (1-x)^{1-x} \ dx, ...

**1**

vote

**0**answers

56 views

### Inverse Laplace Transform involving irrational powers

Could anybody please suggest a reference or a possible solution how to invert the Laplace transform of
$e^{-(s^{\alpha}+\lambda)^{\beta}}$, where $0<\alpha<1$ and $0<\beta<1$.

**0**

votes

**1**answer

147 views

### Subspaces of $H^{\infty}(\mathbb{D})$ which contains a nontrivial weak* closed subalgebra

Let $H^{\infty}(\mathbb{D})$ denotes the Banach space of bounded holomorphic functions in the unit disc. Consider the weak* topology on $L^{\infty}(\mathbb{T})$
that it inherits as the dual of ...

**11**

votes

**1**answer

187 views

### Is there a proof of the uniformization theorem using circle packing?

In this paper: http://www.dm.unipi.it/~benedett/rodin-sullivan.pdf
Rodin and Sullivan show that circle packings converge to the Riemann map. Later, Scharmm and He found another proof of the same ...

**2**

votes

**1**answer

150 views

### Control of a meromorphic function according to distance between its zeros

My question is rather philosophical : can a meromorphic function of normalized norm with simple zeros on the flat torus stay close to zero on a large set when its zeros are far from each other ?
The ...

**2**

votes

**2**answers

139 views

### Asymptotics of the derivatives of analytic functions

Are there sources that treat questions like the following ones?
Suppose that $f\colon\mathbb{C}\to\mathbb{C}$ is an entire function such that $f(x)$ is real for all real $x$ and $f(x)\sim1/x$ as ...

**2**

votes

**2**answers

297 views

### H. Cartan's “Variétés analytiques complexes et cohomologie”?

Does anyone know where I might find an online version (for free or purchase, translated or in french) of this paper by Henri Cartan from 1953? I know it was published in Colloque sur les fonctions de ...

**2**

votes

**0**answers

39 views

### A conformal mapping onto a region bounded by convex contours (Ahlfors)

I want to solve the following exercise (from Ahlfors' text, page 261). I've tried posting it on MSE, and placed a generous bounty, but I couldn't get any answers there.
*3. Using Ex. 2, show that ...

**6**

votes

**1**answer

134 views

### Factorization of conformal maps between annuli

Consider two doubly-connected open subsets $A$ and $A'$ of the Riemann sphere. We assume these two domains to be of same modulus (the moduli space being one real parameter), i.e. we assume that there ...

**8**

votes

**2**answers

259 views

### A specific linear differential equation on $\mathbb{C}-\{0,1\}$ whose monodromy group represents the fundamental group of $\mathbb{C}-\{0,1\}$

Is there a linear differential equation on $\mathbb{C}$ with singularities at $0$ and $1$ whose monodromy group represents the fundamental group of $\mathbb{C}-\{0,1\}$? If so, can someone give a ...

**18**

votes

**1**answer

292 views

### Positivity of coefficients of the inverse of a certain power series

Consider the unique formal power series $g(z)$ with $g(0)=0$ and $g'(0)=1$ satisfying the equation
$$
g(z)-g(z)^8+g(z)^{15}=z,
$$
that is the inverse of
$$
z-z^8+z^{15}
$$
in the group of formal ...

**5**

votes

**1**answer

145 views

### On a Sum of Gamma Functions

I am working on a problem where the following sum appears:
$$F(s, t)=\frac{1}{\Gamma(1+2\alpha)}\sum_{n=0}^{\infty}{\frac{s^{n} ...

**8**

votes

**1**answer

325 views

### Removing singularities in generating functions

This is a problem about the practicalities of removing singularities in multivariable complex functions.
In trying to derive the generating function (in two variables) for a certain problem in ...

**4**

votes

**0**answers

78 views

### Concluding that the Poisson kernel is indeed the Cauchy distribution?

See here.
Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. We see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, ...

**2**

votes

**0**answers

138 views

### Continued Fraction: Please prove $\frac{1}{e \gamma (x+1,1)}=x+\frac{1}{x+1+\frac{2}{x+2+\frac{3}{x+3+\frac{4}{\dots}}}}$ [closed]

I have been playing around with Mathematica and continued fractions and I noticed something.
ContinuedFractionK[n, n + x, {n, 1, Infinity}] ==-x + 1/(E Gamma[1 + x] - E Gamma[1 + x, 1])==-x + 1/(E ...

**2**

votes

**0**answers

75 views

### Poisson kernel, follow-up question, follows that process $\left\{e^{i\theta X_t - \theta Y_t}\right\}$ is a martingale? [closed]

See here.
Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. For any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, ...

**7**

votes

**4**answers

665 views

### Is an entire function, with nowhere vanishing derivative, always a covering map?

Assume that $f:\mathbb C\to\mathbb C$ is entire, and also that $f'(z)\ne 0$, for all $z\in\mathbb C$. Does that imply that $f$ is a covering map of $f[\mathbb C]$?
Clearly, $f$ is a local ...

**7**

votes

**2**answers

312 views

### What are some important papers that use complex analytic techniques to get good bounds?

The motivation behind this question is somewhat similar to that of the tricky project launched by Gowers et al, but is certainly a specialization. My work tends to rely on both exact formulae and ...

**7**

votes

**2**answers

193 views

### How to formalize the *loci of equal arg($\zeta(s)$)* (“isogones”) in the near of a nontrivial root

(This is an extension and specification of a question which I initially asked in MSE having now one comment (which I could not yet digest completely) and which I also detailed further (after working ...

**5**

votes

**1**answer

238 views

### harmonic extension of a curve by different parametrization

Let us consider a curve $\gamma :S^1 \rightarrow \mathbb{R}^3$ (or even a planar convex one if it simplifies). Then I look to the harmonic extension to the disc $h:\mathbb{D}\rightarrow \mathbb{R}^3$ ...

**2**

votes

**0**answers

173 views

### Can one integrate around a branch-cut?

How meaningful is it to try to integrate around the branch-cut of a function?
For example lets say I have the function $\log(z^2+a^2)$ for $a>0$ and I choose my branch-cuts to be starting at $\pm ...

**1**

vote

**1**answer

170 views

### Stone-Weierstrass Theorem, polynomial interpolation, divided difference in complex plane

Setting:
Let $\Gamma$ be a simple smooth($C^\infty$) curve in $\mathbb{C}$ parametrized by the injective map $\gamma:[0,1] \to \mathbb{C}$.
Assume $f$ is a function defined on $\Gamma$ s.t. $f$ is ...

**3**

votes

**0**answers

127 views

### polynomial relations between modular functions

$\newcommand{\Qbar}{\overline{\mathbb{Q}}}$
We define a modular function to be a meromorphic modular form of weight 0 for some subgroup (not necessarily congruence) $\Gamma\le\text{SL}_2(\mathbb{Z})$ ...

**1**

vote

**0**answers

65 views

### Construction of homogeneous Siegel domain from j-algebra

I am reading bounded homogeneous domain from Piatetski-Shapiro's
book ``Automorphic functions and the geometry of classical domains''
and have questions on how to construct homogeneous Siegel domain
...