**7**

votes

**3**answers

827 views

### generalisation of Cauchy-Riemann equations to 3D

Hi, harmonicity in 2d is preserved under mappings that satisfy the Cauchy-Riemann equations.
What about 3D? What conditions should a mapping satisfy to preserve harmonicity?
is there a general ...

**3**

votes

**1**answer

88 views

### Xi Function on Critical Strip - Mellin Transform

Story
I'm trying to prove following identity
$$\int_0^\infty \frac{\Xi(t)}{t^2 + \frac{1}{4}} \cos(xt) dt = \frac{1}{2} \pi (e^{\frac{1}{2}x} - 2e^{-\frac{1}{2}x} \psi(e^{-2x}))$$
where
$$\psi(x)=\...

**10**

votes

**6**answers

691 views

### Identities and inequalities in analysis and probability

Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of course,...

**0**

votes

**0**answers

59 views

### On semi-complete ring K[X_1,X_2,…,X_∞]] and Popescu theorem

Let $P_n \colon= K[X_1,...,X_n]$ be a $n$-variables polynomial ring. We define 'semi-complete' polynomial ring $P_{\infty}$ by the following$\colon$
$P_{\infty} = K[X_1,...,X_\infty]] \colon = \...

**2**

votes

**0**answers

111 views

### Parametric normalized Fermat curves (Fermat functions)

The normalized Fermat curve is $X^n+Y^n=1$. We have of course infinite possibilities for parametrisation, but the periodicity is a special characteristic here.
E.g. $\cos^2x+\sin^2x=1$ has ...

**1**

vote

**0**answers

17 views

### Generalizing an expected increase in autocorrelation near a bifurcation point to a system of ODE

Near a bifurcation point, a stochastically forced dynamical system should show an increase in autocorrelation and variance. This is due to critical slowing (a loss in resilience to perturbations). ...

**-1**

votes

**0**answers

24 views

### Complex fixed points on the bifurcation diagrams [closed]

I'm working with bifurcation diagrams, an extesion that is being made of them is the determination of complex fixed points in addition to the real fixed points.
Given a dynamic system (e.g. an ...

**1**

vote

**0**answers

104 views

### Gravitational instantons metric (change variables)

I'm trying to understand the paper by Hitchin called: ''Polygons and gravitons". I'm stuck at page 471. At this point, he does some computations and obtains a metric:
$$
\gamma dz d\bar{z}+\gamma^{...

**1**

vote

**1**answer

231 views

### A Geometric proof of the Gauss Lucas theorem

Motivated by a geometric proof of the Fundamental Theorem of Algebra we ask:
Is there a geometric proof for the Gauss-Lucas theorem ?Since we are working on a half plane, can one imagine a possible ...

**1**

vote

**1**answer

45 views

### On the limit set of eigenvalues of banded Toeplitz Hessenberg matrices

Let $T_{n}(b)$ be the $n\times n$ Toeplitz matrix determined by the symbol
$$
b(z)=\frac{1}{z}+\sum_{j=0}^{k}a_{j}z^{j}
$$
where $k\in\mathbb{N}$ and $a_{0},\dots,a_{k}\in\mathbb{R}$, $a_{k}\neq0$. ...

**2**

votes

**1**answer

276 views

### Half spaces free of roots of a given polynomial

I thank Loic Teyssier and Emil Jerabek who helped me to revise the two previous version
This question is motivated by the following fact in complex variable:(I learned this fact from the book of ...

**0**

votes

**1**answer

56 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

**1**answer

162 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

**1**answer

139 views

### Complete subring of F_p[[X]]

Pointed out on famous disbelief, I know now that there is an embedding
$\iota_n \colon {\Bbb F}_p[[T_1,...,T_n]] \hookrightarrow {\Bbb F}_p[[X,Y]]$
for any $n \leq \infty$. Then I would like to ask ...

**4**

votes

**1**answer

237 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

**0**answers

52 views

### Coherence of subrings of K[[X,Y]]

Let $K[[X,Y]]$ be a two-variables formal power series ring over a field $K$. Consider a sub-ring $\iota \colon A \subset K[[X,Y]]$.
Q. Is A coherent? $\quad$ Or is it automatic that $\iota$ is ...

**2**

votes

**1**answer

58 views

### Uniform Mahler Measure Lower Bound

I'm working on a problem in multiplicative ergodic theory, and Mahler measure has just made another appearance. I am looking for a uniform lower bound on Mahler measure over all polynomials of fixed ...

**3**

votes

**1**answer

110 views

### Analytic Combinatorics: upper bound for sum of absolute values of two complex functions: $|z f'(z)| + |2 f(z) - zf'(z)| \leq 2f(|z|)$

Let $f \colon \mathbb C \to \mathbb C$ be a complex-valued analytic function with non-negative coefficients of Taylor series at 0 (suppose that radius of convergence is $+\infty$ for simplicity):
$$
...

**2**

votes

**1**answer

241 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

**2**answers

106 views

### Bieberbach-type bound for bounded univalent functions

Suppose $f: \mathbb{D}\to \mathbb{C}$ is a univalent function with $$f(z)=z+a_2z^2+a_3z^3+\cdots.$$ The Bieberbach conjecture/de Branges' theorem asserts that $|a_n|\leq n$ with equality for the Koebe ...

**7**

votes

**1**answer

335 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
$...

**7**

votes

**1**answer

287 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

**1**answer

85 views

### Neighborhoods with proper multiplication

The following question was originally asked here, by C. Dubussy: http://math.stackexchange.com/questions/1802111/neighbourhoods-with-proper-multiplication
Assume we have two closed subsets $F$ and $G$...

**6**

votes

**1**answer

483 views

### Is this a semi algebraic set?

Put $$A=\{(a_{0},a_{1},\ldots,a_{n}) \in \mathbb{C}^{n+1}\mid p(z)=a_{0}+a_{1}z+\ldots a_{n}z^{n} \;\;\text{is a one-to one function on the unit disc} \{z\in \mathbb{C} \mid |z|\leq 1\}$$
Is $\{(...

**1**

vote

**0**answers

46 views

### Does cutting off the taylor expansion of e^x always give an irreducible polynomial? [duplicate]

I am talking of the polynomials:
$P_n(x)$ = $1+x..+x^n/n!$
I've tested this for the first 10 values and it seems so. I know this might be random but I've got a hunch that there's something deeper ...

**1**

vote

**0**answers

112 views

### Can an algebraic function be zero both at $z=0$ and at its leading singularity?

Apologies for asking possibly strange questions, but I am just a poor computer scientist trying to understand a mathematical paper on singularity analysis of algebraic functions that is apparently not ...

**1**

vote

**0**answers

91 views

### Finding stable ideals of $\mathbb{F}_3[[X,S]]$ by group action

Let $k > 1$ be a positive integer and define the action $\sigma_k$ on $\mathbb{F}_3[[X,S]]$ by:
$\sigma_k: X \mapsto X + S + X^k$
$\sigma_k: S \mapsto S + S^3$.
Conjecture: There exists a ...

**6**

votes

**0**answers

184 views

### Is the space of holomorphic maps a manifold

To be more specific:
Let $Q\subset\mathbb{C}$ be a Lipschitz bounded domain, and $V$ is a compact complex manifold without boundary. Consider the set of holomorphic maps $f:Q\rightarrow V$, and $f\in ...

**2**

votes

**0**answers

130 views

### system of complex number equations

Let $a_1,a_2,a_3,a_4\in \mathbb{C}$ be distinct such that
$$a_1^3+a_2^3+a_3^3+a_4^3=0$$
$$(1+|a_1|^2)a_1^2+(1+|a_2|^2)a_2^2+(1+|a_3|^2)a_3^2+(1+|a_4|^2)a_4^2=0$$
$$(1+2|a_1|^2+2|a_1|^4)a_1+(1+2|a_2|^...

**6**

votes

**1**answer

289 views

### Is polynomial convexity a topological invariant?

Is the property of being polynomially convex a topological invariant?
In other words, let $M$ and $N$ be two homeomorphic, compact subsets of $n$-dimensional complex Euclidean space, and assume in ...

**8**

votes

**4**answers

765 views

### A learning roadmap to the Schramm-Loewner evolution (SLE) for the complex analyst

I would like some good references to learn about the Schramm-Loewner evolution (SLE), for a complex analyst with no background in probability.
A quick google search gave a lot of references on SLE ...

**4**

votes

**1**answer

65 views

### Integral Expression in Complex Dynamics

Let $\phi\in \mathbb{C}(z)$ be a degree $d\geq 2$ rational map, which we can write as $\phi = \frac{f}{g}$ for $f,g\in \mathbb{C}[z]$. Let $\omega_{FS}$ denote the Fubini-Study form on $\mathbb{P}^1(\...

**3**

votes

**1**answer

108 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 $s=\...

**1**

vote

**0**answers

55 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

**2**answers

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 ...

**3**

votes

**4**answers

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 ...

**-4**

votes

**1**answer

283 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 ...

**2**

votes

**2**answers

160 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**

votes

**0**answers

152 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

**2**answers

215 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

**2**answers

459 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**

votes

**0**answers

80 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

**0**answers

77 views

### Octahedron and System of trigonometric equations

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

**1**

vote

**0**answers

97 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 $\mathbb{...

**0**

votes

**0**answers

35 views

### Inverse Mellin transform of ratio of gamma functions

Any pointers on how to solve the inverse Mellin transform below:

**1**

vote

**2**answers

198 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

**3**answers

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**

votes

**1**answer

82 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**

vote

**0**answers

79 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 \overline{z}}-z\frac{\...