**-2**

votes

**0**answers

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

**1**answer

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**

vote

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**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 ...

**4**

votes

**1**answer

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**

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 ...

**2**

votes

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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**

votes

**0**answers

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

**2**answers

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

**3**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 ...

**0**

votes

**0**answers

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

**1**answer

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**

vote

**0**answers

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**

votes

**0**answers

32 views

### Inverse Mellin transform of ratio of gamma functions

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

**1**

vote

**2**answers

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

**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

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**

vote

**0**answers

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

**46**answers

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

**3**answers

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

**1**answer

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

**0**answers

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

**3**answers

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**

votes

**1**answer

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**

votes

**3**answers

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**

votes

**3**answers

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**

vote

**0**answers

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**

votes

**1**answer

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

**2**answers

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**

votes

**4**answers

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**

votes

**1**answer

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

**1**answer

263 views

### Two similar integrals

Let $n$ be a given even positive integer. We have the following integral
\begin{align}
...

**7**

votes

**1**answer

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**

vote

**0**answers

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**

votes

**0**answers

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

**4**answers

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**

vote

**0**answers

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**

vote

**1**answer

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**

vote

**1**answer

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**

vote

**1**answer

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

**0**answers

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**

votes

**0**answers

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

**1**answer

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**

votes

**2**answers

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 ...