All Questions
3,561 questions
5
votes
3
answers
314
views
fixed points of quadratic iteration
Consider the well-known iteration $f:z\to z^2 + c,$ and consider the values of $c$ for which $0$ is a periodic point. Experiment shows that most such values of $c$ (about $480$ out of $512$ for period ...
- 96.4k
3
votes
1
answer
304
views
Relationship between volume and area
Let $\mu(z) dV_n$ be a measure in $\mathbb{C} ^n$.
Let $B_n(r) := \{z \mid \|z\| < r\}$ be the ball of radius $r$ in $\mathbb C^n$, and $\partial B_n(r) $ be the corresponding sphere.
In $\mathbb{C}...
- 33
6
votes
0
answers
228
views
All complex surfaces embed into a common complex manifold
Is there a closed complex manifold into which every closed complex surface embeds?
user164740
2
votes
1
answer
192
views
Existence of a distinguished continuous version of the logarithm of a continuous function
Let $E$ be a $\mathbb R$-Banach space and $\varphi\in C^0(E,\mathbb C\setminus\{0\})$ with $\varphi(0)=1$.
I want to show that there is an unique $\psi\in C^0(E,\mathbb C)$ with $\psi(0)=0$ and $$\...
- 167
1
vote
0
answers
160
views
Order of vanishing of an analytic function along its vanishing locus
Suppose $F: {\mathbb C}^n \to {\mathbb C}^k$ is a holomorphic/analytic function with vanishing locus $V_F = F^{-1}(0)$. Can one prove that there are positive real numbers $C>0$ and $\delta>0$ ...
- 803
3
votes
0
answers
68
views
Analogue of Carlson's theorem for poles in the upperhalf plane?
Let $f:\mathbb{H}\to \mathbb{C}$ be a holomorphic function on the upper half plane. Even it is not defined on the real line, we will define $\mathrm{ord}_{z=z_0}f(z)$ to be the unique value $\xi\in\...
- 2,902
17
votes
3
answers
767
views
Can all $L^2$ holomorphic functions on a domain vanish at a particular point?
Let $\Omega \subset \mathbb{C}^n$. Is it possible that there is a point $p \in \Omega$ such that every $f \in A^2(\Omega) = L^2(\Omega) \cap \mathcal{O}(\Omega)$ has a zero at $p$? The space $A^2(\...
- 12.1k
1
vote
1
answer
324
views
injective holomorphic mapping between unit disk and unit polydisk
In $\mathbb{C}^n,\ n\geq 2$, there is no bijection between unit disk $B^n(0,1)$ and unit polydisk $P^n(0,1)$. But if we wish to find injective holomorphic mapping from unit disk to polydisk(whose ...
- 95
8
votes
1
answer
594
views
Reference for flatness in complex-analytic geometry
What is a good reference for flat morphisms of complex-analytic spaces? (The book by Grauert and Remmert doesn't treat them).
Topics I'm interested in: openness of flat maps, descent for coherent ...
user95222
1
vote
1
answer
305
views
Map from local systems to holomorphic line bundles on a curve
Let $X$ be a Riemann surface of genus $g > 0$. Let $S$ denote the set of local systems (locally constant sheaves) on $X$ with fiber $\mathbb{C}$. $S$ is in natural bijection with $H^1(X, \underline{...
- 1,046
4
votes
2
answers
912
views
Residue for the generating function of the Euler totient function
Let $\varphi$ be the Euler totient function, and let us define the function $f(z)$ by the series
$$
f(z) := \sum_{n=1}^{\infty} \varphi(n) z^n
$$
Since $0\le \varphi(n)\le n$, I believe this gives a ...
3
votes
1
answer
220
views
Reference request: The transform of a bounded random variable has a zero in the complex plane
Together with coauthors I'm working on a paper where we use the following Proposition:
If a real-valued random variable $X$ has bounded support, then except in the trivial case that $X$ has all ...
- 5,492
8
votes
3
answers
841
views
Holomorphic function with a.e. vanishing radial boundary limits
Hello everybody.
I'm looking for an "easy" example of a (non-zero) holomorphic function $f$ with almost everywhere vanishing radial boundary limits: $\lim\limits_{r \rightarrow 1-} f(re^{i\phi})=0$.
...
- 742
4
votes
1
answer
410
views
A continuous function on the disk without non-tangential limits
By Fatou's theorem every bounded holomorphic function on the unit disk has non-tangential limits almost everywhere on the unit circle $\mathbb T$.
Is there an explicit example of a bounded continuous ...
- 687
10
votes
1
answer
468
views
Bounded holomorphic functions on a Riemann surface separating points
Let $R$ be a Riemann surface that admits a non-constant bounded holomorphic function. Then is it true that any two points of $R$ can be separated by a bounded holomorphic function? This is easy to see ...
- 1,169
1
vote
1
answer
303
views
Cauchy's Integral with quadratic exponential term
As I was studying the Cauchy's integral formula, I tried to do the integral:
\begin{equation}
I = \int\limits_{-\infty}^{\infty} \frac{1}{x - a} e^{(i A x^2 + i B x)} dx
\end{equation}
with $A>0, ...
8
votes
2
answers
330
views
Equivalence of definitions of quasiconformal surfaces?
I have been reading John H. Hubbard's book Teichmüller Theory vol. 1 and I am a little bit concerned with his definition of quasiconformal surface.
Definition: A quasiconformal surface $S$ is a ...
- 298
32
votes
3
answers
2k
views
How is the Julia set of $fg$ related to the Julia set of $gf$?
Let $f$ and $g$ be complex rational functions (of degree $\geq 2$ if that helps). What can be said about the relationship between $J(fg)$ and $J(gf)$, the Julia sets of the composite functions $f \...
- 27.7k
0
votes
1
answer
425
views
Are anti-linear maps/semi-linear, such as conjugations, linear in other almost complex structures?
I have asked this on mse, but I did not get any responses even after a bounty.
I started studying the book of Daniel Huybrechts, Complex Geometry An Introduction. I tried studying backwards as much ...
- 247
3
votes
0
answers
370
views
When are two complex Tori biholomorphic
Let $g \ge 1$ be a natural number and $\mathbb{C}^g$ complex vector space which
is isomorphic to $\mathbb{R}^{2g}$ is real vector space.
An additive subgroup $\Gamma \subset \mathbb{C}^g$ is called a
...
- 6,016
5
votes
0
answers
245
views
Dimension of highest discriminants of a morphism
Let $f: X\to Y$ be a flat morphism between smooth complex affine varieties. Let $Z$ be the closed set of most singular points of $f$ (in the sense: $p$ is a most singular point of $f$ if the tangent ...
- 1,081
5
votes
1
answer
290
views
Overconcentration of Poles on the Circle of Convergence of a Power Series with Bounded Coefficients
Let $V$ be an arbitrary set of infinitely many positive integers, and let: $$\varsigma_{V}\left(z\right)\overset{\textrm{def}}{=}\sum_{v\in V}z^{v}$$ Let $T_{V}$ denote the set of all $t\in\left[0,1\...
- 1,284
5
votes
1
answer
898
views
Are there enough meromorphic functions on a compact analytic manifold?
Let $X$ be a compact complex analytic manifold, $D\subset X$ an irreducible smooth divisor, given as zeroes of a global meromorphic function $f\in {\mathfrak M} (X)$. Are there enough other ...
- 12.4k
1
vote
0
answers
38
views
Solving an equation containing Laplace transform
Consider the equation
\begin{equation}
\frac{f(p)}{f(s_{1})}\mathcal{L}(y)(s_{1})+\frac{g(p)}{g(s_{2})}\mathcal{L}%
(y)(s_{2})=\mathcal{L(}y)\mathbf{(}p),
\end{equation}
where $\mathcal{L}$ is the ...
- 47
0
votes
0
answers
83
views
The loss of double periodicity (ellipticity)
Consider a meromorphic function $f(\mathfrak{a}_1, \mathfrak{a}_2)$, such that
$$
\begin{align}
f(\mathfrak{a}_1, \mathfrak{a}_2) = f (\mathfrak{a}_1 + 1, \mathfrak{a}_2) = f(\mathfrak{a}_1 + \tau, \...
- 857
5
votes
0
answers
95
views
Convergence of Hahn series
Enumerate $\Bbb{Q}^+$ with $\Bbb{Z}^+$ by a bijiective map $f:\Bbb{Z}^+ \rightarrow \Bbb{Q}^+$. Consider the Hahn series: $$P_f(x)=\sum_{n=1}^{+\infty}c_nx^{f(n)}$$ where $c_n \in \Bbb{C}$, $x \in \...
- 1,543
8
votes
2
answers
2k
views
Examples of analytic functions to motivate a first course in complex variables
[Changed title as a plea to re-open the question.]
If one is to motivate a course in complex variables, what specific analytic (holomorphic/meromorphic) function of one variable would you cite as an ...
1
vote
1
answer
476
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:
$$
\gamma dz d\bar{z}+\gamma^{-...
- 61
1
vote
1
answer
674
views
Entire even functions of order 1 have infinitely many zeros?
Let $f$ be an entire even function of order 1 such that $f(0)\neq 0$. Does $f$ have infinitely many zeros?
11
votes
3
answers
900
views
How many isolated roots can a polynomial in $z$ and $\overline{z}$ have?
This is probably well-known by I can't find it now online. My guess is that if the degree is $n$ then it's $2n$ but it's just a hunch.
EDIT:
This is an edited version. Before I asked about roots ...
- 6,962
1
vote
0
answers
124
views
Context for this discrete Cauchy integral formula
Notation: I will use the following conventions for discrete Fourier transforms (DFT) and discrete time Fourier transforms (DTFT):
$$\mathcal{D}_N[x_j](k) := \sum_{j=0}^{N-1} e^{-2\pi i j k} x_j$$
$$\...
- 956
5
votes
1
answer
339
views
A variant of Cauchy-type functional equation conjecture
Let $f:\mathbb{C}\to \mathbb{C}$ be a complex function such that
$$|f(x-y)|=|f(x)-f(y)|,\qquad x,y\in\mathbb{C}.$$
Is it true that $$f(x+y)=f(x)+f(y),\qquad x,y\in\mathbb{C}?$$
The answer is ...
- 4,280
12
votes
11
answers
2k
views
Giving a math talk with no blackboard or projector
I need to give a math talk to a group of undergraduates. I am asking for advice because this talk will take place at a department picnic and there will be no blackboard or projector. I would like to ...
-3
votes
2
answers
225
views
Zeroes of linear combination of sines [closed]
Let us consider $$f(z):=\sum\limits_{j=1}^{j=n}a_j\sin(\lambda_jz) $$
where all $a_j$ and $\lambda_j$ (of course, $\lambda_j$ are distinct) are real numbers and $n \in \mathbb{Z},\, n \ge 3$. The ...
- 3,486
1
vote
0
answers
203
views
Construction of weight function to satisfy condition on given functional
Consider the following function :
$$F(z) = \omega(z){\sin^2\left(\frac{c\Gamma(z)}{z}\right)}$$
Here, $\omega(z)$ is a weight we are going to consider
The following two conditions should meet for $\...
- 375
27
votes
3
answers
2k
views
Kasteleyn's formula for domino tilings generalized?
It seems a marvel when a bunch of irrational numbers "conspire" to become rational, even better an integer. An elementary example is $\prod_{j=1}^n4\cos^2\left(\pi j/(2n+1)\right)=1$.
Kasteleyn's ...
- 43.2k
6
votes
1
answer
4k
views
Examples of separable ordinary differential equations in economics
I'm currently teaching an integral calculus course for business students, and we're just about to discuss differential equations. They've worked hard, and I'd like to reward them with some economic ...
- 1,665
2
votes
0
answers
119
views
An analytic function, asymptotically expandable in a Dirichlet series, is the sum of this series
Let there be a function $F(s)$ that is analytic in some half-plane $\sigma>\sigma_0$ (where $s=\sigma + it $). Let the function $F(s)$ have an asymptotic expansion of the form $F(s)\sim\sum\limits_{...
33
votes
1
answer
2k
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 ...
- 7,426
0
votes
1
answer
124
views
Image of transcendental meromorphic functions
Let $f$ be a trancendental meromorphic function such that $f'(z) \ne 0$ for all $z \in \mathbb{C}$. Let $\Pi$ be the stereoprojection map from the north pole on the unit sphere. My question is the ...
- 1,350
18
votes
3
answers
2k
views
Does Riemann map depend continuously on the domain?
I've always taken this for granted until recently:
In the simplest case, given Jordan curve $C \subseteq \mathbb{C}$ containing a neighborhood of $\bar{0}$ in its interior. Given parametrizations $\...
- 375
3
votes
1
answer
174
views
A question on preimage of a locally injective meromorphic function
Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a meromorphic function such that $f'(z) \ne 0$ for all $z \in \mathbb{C}$. Let $\Gamma \subset \mathbb{C}$ be a curve which has no self intersections. If ...
- 1,350
5
votes
1
answer
159
views
$C^j$-topology considered by Greene and Krantz
My question is about the $C^j$ topology used by Greene and Krantz in their paper "Deformations of Complex Structures, Estimates for the $\bar{\partial}$-equation, and stability of the Bergman ...
- 113
2
votes
1
answer
120
views
Adjoint operator
Assume that $$\left<f,g\right>_R=\Re \int_U f(z) \overline{g(z)} \, dx \, dy, f,g \in L^2(U),$$ where $U$ is the unit disk and assume that $A: L^2(U)\to L^2(U)$ is a real-linear operator. Assume ...
- 719
2
votes
1
answer
232
views
Continuous extensions of Riemann mappings
Let $K$ be a compact set in $\mathbb C$ without interior. Suppose, additionally, that $K$ is a retract (or equivalently $K$ connected, $K$ locally connected and $\mathbb C\setminus K$ connected). ...
- 687
2
votes
1
answer
216
views
A generalization of polynomial algebra on a Riemann surface
Let $M$ be a $1$-dimensional complex manifold. Let $A$ be the space of all holomorphic functions $f:M\to \mathbb{C}$ such that either $f$ is a constant function or every level set $f^{-1}(c)$ is a ...
- 366
3
votes
1
answer
336
views
Conformal mapping between two right-angled triangles
I want to derive a conformal mapping $f\!:\!A\!\to\! B$ where $A=\{ (x,y)\ |\ x\!\in\![0,1]\ \text{ and }\ 0 \leq y \leq x \}$ and $B=\{ (x,y)\ |\ x\!\in\![0,1]\ \text{ and }\ 0 \leq y \leq \frac{...
- 167
10
votes
2
answers
4k
views
Power series with funny behavior at the boundary
Consider a power series
$$
\sum_{n=0}^{\infty}a_nz^n
$$
where $a_n$ and $z$ are complex numbers. There is radius $R$ of convergence. Let us assume that is a positive real number. It is well known that ...
- 2,384
11
votes
1
answer
1k
views
Dual of the space of all bounded holomorphic functions
Let $\mathbb{B}$ be the open unit ball in $\mathbb{C}^n, n\geq 1$ and let $H^\infty (\mathbb{B})$ be the space of all bounded holomorphic functions on $\mathbb{B}$. It is well known that $H^\infty (\...
- 211
9
votes
1
answer
322
views
Notational question about quadratic differentials in Strebel's book "Quadratic differentials"
In Kurt Strebel's book "Quadratic Differentials", in Chapter 2, $\S4$, he begins by saying:
"Every analytic function $\varphi$ is a domain $G$ of the $z$-plane defines, in a natural way, a field of ...
- 7,547