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

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3
votes
1answer
90 views

What is the correct generalization of the Wirtinger derivatives to arbitrary Clifford algebras?

In the complex numbers setting, the two Wirtinger derivatives are defined as: $\frac{\partial}{\partial z}= \frac{1}{2}\left( \frac{\partial}{\partial x} - i \frac{\partial}{\partial y} ...
47
votes
9answers
8k views

Why do functions in complex analysis behave so well? (as opposed to functions in real analysis)

Complex analytic functions show rigid behavior while real-valued smooth functions are flexible. Why is this the case?
3
votes
0answers
186 views

Inverse problem for negative moments

Let $D$ be a bounded simply connected domain in the plane, bounded by a smooth closed curve $\partial D$. Moreover$D$ contains the origin. Assume that all negative complex moments w.r.t arc-length ...
5
votes
1answer
160 views

Analytic continuation of the double sum $\sum_{n,m\ge0}x^ny^mt^{nm}$

Define function $f(x,y,t)$ as the analytic continuation of the series $$f(x,y,t)=\sum_{n,m\ge0}x^ny^mt^{nm}$$ This series definitely converges when all the arguments are small enough. My goal is to ...
2
votes
1answer
140 views

Strange limit problem involving $\binom{z}{n} e^{-xn\log n}$ with $z \in \mathbb{C}$

Is the following limit result correct: $$\lim\limits_{x \to 0^{+}}\sum\limits_{n=0}^{\infty} \binom{z}{n} e^{-xn\log n} = 2^{z}$$ where, $z \in \mathbb{C}$, and the notation $\displaystyle ...
25
votes
1answer
713 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 ...
1
vote
1answer
103 views

When is a homogeneous polynomial an inner function on the unit torus?

What is the necessary and sufficient condition(s) for a homogeneous polynomial on torus(bidisk) to be an inner function? More precisely I want to know when absolute value of a homogeneous polynomial ...
0
votes
1answer
57 views

A Gaussian integral over complex variables by a defined Green's function for a Gaussian ensemble of random matrix

We construct an $N\times N$ matrix $J$ whose elements are drawn from Gaussian distribution with zero mean and variance $\frac{1}{N}$. Since we want to have different variances for different columns, ...
9
votes
3answers
404 views

Complex proof of $B(a,b)=\Gamma(a)\Gamma(b)/\Gamma(a+b)$

It is a question in spirit of this one. Is there a way to prove Euler's formula $$ \int_0^1 x^{a-1}(1-x)^{b-1}dx=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)} $$ using contour integration (and maybe ...
0
votes
0answers
58 views

Existence of analytic continuation of $f(z)=\sum{n^{\alpha}} z^n$ for fractional $\alpha$

The power series $f(z)=\sum_{n \ge1}{n^{\alpha}} \cdot z^n$ has radius of convergence 1. For $\alpha \in \mathbb{N}$ it is easy to see that $f$ permits an analytic continuation to $\mathbb{C} ...
-1
votes
0answers
11 views

How can i get real analog of complex function? [migrated]

I have a function: sin(wt-jT) (1.1), where j - complex number I transform it to function with real arguments: ...
1
vote
2answers
112 views

Examples of pluripolar sets

I have a very basic question on pluripolar sets. First remind their definition. Let $\Omega\subset \mathbb{C}^n$ be a domain. A subset $E\subset \Omega$ is called pluripolar if there exists a ...
1
vote
0answers
49 views

Inverse Laplace transform of a non-negative function

Consider an entire function $f$, which is real for real arguments and satisfies $f(s)\geq 0$ for all $s\in\mathbb{R}$. Furthermore, assume this function is a Laplace transform, $$ f(s)=\int_0^\infty ...
1
vote
0answers
93 views

Does Feuter regularity imply derivability in all directions?

The standard type of regularity in Clifford Calculus is the one introduced by Feuter, namely: a function is Feuter regular iff it is in the zero set of the Clifford-Dirac operator $D= ...
30
votes
3answers
927 views

Polynomials with the same values set on the unit circle

Assume that $P(z)$, $Q(z)$ are complex polynomials such that $P(S)=Q(S)$, where $S=\{z\colon |z|=1\}$ (equality is understood in the sense of sets, but I do not know the answer even for multisets). ...
9
votes
2answers
319 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
0answers
29 views

Is there an analytic function such that [migrated]

can you help me understand the identity theorem. The question is: does it exist an analytic function such that: $f(\frac{1}{n})=f(\frac{-1}{n})=\frac{1}{n^2}$ or ...
12
votes
2answers
1k views

The Cauchy–Riemann equations and analyticity

I would be interested to learn if the following generalization of the classical Looman-Menchoff theorem is true. Assume that the function $f=u+iv$, defined on a domain $D\subset\mathbb{C}$, is ...
3
votes
0answers
145 views

Quadrature domains for arc length

Is ellipse a quadrature domain for arc-length? More precisely does there exist points $z_1,\cdots,z_n$ inside an ellipse $E$ and non zero constants $c_1,\cdots,c_n$ such that $$\int ...
-1
votes
0answers
25 views

state-of-art numerical contour (complex) integration method when contour is square and available values are evenly spaced

What is current state-of-art for numerical contour integration method (for $f(z)$ with $z$ being complex number and $f$ complex-valued) when contour is square on complex plane, and one only has ...
29
votes
7answers
6k views

Interpreting the Famous Five equation [closed]

$$e^{\pi i} + 1 = 0$$ I have been searching for a convincing interpretation of this. I understand how it comes about but what is it that it is telling us? Best that I can figure out is that it just ...
12
votes
4answers
878 views

Complex evaluation of a classical (real) integral

There are several ways to compute the classical integral $$ \int_{\mathbb R}e^{-x^2}dx=\sqrt{\pi}. $$ Probably, best known are (1) squaring the integral with subsequent change of (now two) variables ...
2
votes
0answers
43 views

Holomorphic solution to SDE

Consider the SDE $dZ_t = \mu(t,x) d_t + \sigma(t,x) dW_t$. Are there any known (necessary and) sufficient conditions on $\sigma(t,x)$ and on $\mu(t,x)$ guaranteeing that $f(T):=\mathbb{E}[\int_0^T Z_t ...
1
vote
0answers
112 views

Harmonic and Primitive Forms on a Kaehler Manifold

Let $M$ be a compact Kaehler manifold, and $p$ a primitive form, which is to say it is contained in the kernel of the adjoint of the Lefschetz operator $L$ associated to the Kaehler form. If $p$ is ...
2
votes
1answer
92 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 $\{r,s\}$ ...
113
votes
46answers
36k 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: ...
12
votes
2answers
974 views

Anything special (historical?) about surface $x\cdot y\cdot z\ +\ x+y+z=0$?

QUESTION I wanted to introduce and develop the complex logarithm from scratch. As the result I've arrived a couple of months ago at the following identity after which the road to complex logarithm is ...
-1
votes
0answers
89 views

Soft question: take complex analysis or cryptology? [migrated]

I am math major junior considering math grad school. I need to decide whether to take complex analysis or cryptology this semester. Complex analysis seems to be a recommended course for people ...
0
votes
0answers
44 views

How to take partial derivative of spherical interpolation of quaternions?

Using the standard definition of quaternionic spherical linear interpolation (slerp): $$ Q(q_0,q_1,t) := q_0(q_0^{-1}q_1)^t, $$ how can I take each partial derivative? Actually, I'm confident how to ...
0
votes
1answer
118 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 $t \to \infty$, ...
1
vote
0answers
88 views

For which integer values of $k$ can we find one solution to the equation $\sum\limits_{n=1}^{n=k} \frac{1}{n^s}=0$ by iteration? [closed]

I am trying to find solutions to the well known equation: $$\sum\limits_{n=1}^{n=k} \frac{1}{n^s}=0$$ Now with this program below I have found that for certain values of the integer $k$ one can find ...
16
votes
1answer
649 views

Are the algebraic numbers dense everywhere on the boundary of the Mandelbrot set?

Let $\mathcal{B}$ denote the boundary of the Mandelbrot set, and let $\overline{\mathbb{Q}}$ denote the algebraic closure of the rationals. Further put $\mathcal{B}_{\overline{\mathbb{Q}}} := ...
0
votes
1answer
143 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 ...
5
votes
1answer
248 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 ...
0
votes
0answers
68 views

A question on evaluation of complex integrals

Is any general relationship between the integral \[ \int_{0}^{1}f(u, \sigma + it)u^{-1 + d}du \] and $f(0, \sigma + it)$ known? I have proved one such result where the main term of the given integral ...
38
votes
4answers
2k views

What is the Krull dimension of the ring of holomorphic functions on a complex manifold?

Consider a connected holomorphic manifold $X$ and its ring of holomorphic functions $\mathcal O(X).$ My general question is simply: in which cases is the Krull dimension $\dim \mathcal O(X)$ known? ...
7
votes
2answers
309 views

regular polygon and constant potential function

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 ...
6
votes
1answer
134 views

Can the potential of a complete Kahler metric be bounded?

Let $X$ be a complex manifold and $\omega$ a Kahler form on $X$. A smooth function $\rho$ is called a potential of $\omega$ if $i\partial\bar\partial\rho=\omega$. By intuition, it seems that $\rho$ ...
2
votes
0answers
85 views

system of complex number equations

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

Behavior of “integer complex number” on computer [closed]

I want to provide software to compute with "integer complex numbers", that live in $\mathbb{Z}\times i \mathbb{Z}$, rather than the $\mathbb{C}$. Some operations are going to give results that are ...
4
votes
0answers
65 views

sums of zero-free entire functions and its siblings on the disk

Can one describe the set $\{e^f+e^g: f, g\in H(C)\}$ in some way? For example, in unital Banach algebras, every element has this form. I am in particular interested in the problem whether the ...
1
vote
0answers
54 views

Octahedron and System of trigonometric equations

Could somebody help me to prove the following? $$\sum_{k=1}^6 \cos(2 \theta_k) (\cos(2\phi_k)-1))=0$$ $$\sum_{k=1}^6 \sin(2 \theta_k) (\cos(2\phi_k)-1))=0$$ $$\sum_{k=1}^6 \cos (\phi_k)=0$$ ...
7
votes
2answers
655 views

Relationships between the roots of an entire function and the roots of its derivative

Hey everyone, I would like to know if anybody could help me find references for the following. Take a suitably well defined entire function $f(x)$ and it's derivative $\tilde{f}(x)$ to which the ...
7
votes
1answer
1k views

Poincaré line bundle

I am being stuck by the proof of the existence of Poincaré line bundle of complex torus in Griffiths-Harris. Here is the question: Let $M$ be a complex torus and $M'$ be the complex torus dual to ...
7
votes
2answers
285 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
0answers
194 views

Picard Fuchs and Lefschetz trace

In Clemen's book "A Scrapbook of Complex Curve Theory", he discusses in Chapter 2 how the infinite sum giving the period of the Legendre curve matches (mod p) the sum giving the number of points over ...
2
votes
1answer
142 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 ...
6
votes
1answer
433 views

Bounds on the derivative of a Riemann map

Let U be a simply connected domain with smooth boundary in the complex plane, and let $\mathbb D$ be the unit disc. Is there a nice sufficient condition for the existence of a biholomorphic map ...
1
vote
2answers
135 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 $a > 0$.
14
votes
1answer
585 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. ...