**3**

votes

**1**answer

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

**9**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**3**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**3**answers

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

**2**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**7**answers

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

**4**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**46**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**4**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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. ...