Holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
0
votes
0answers
32 views
Is it true that the Laplace Transform of a real function with compact support is always entire? [migrated]
Is it true that the Laplace Transform of a real function with compact support is always entire (entire = complex derivative exists on the entire complex plane)?
2
votes
2answers
64 views
Is the Hausdorff dimension $Dim_{H}(J(f))$ of the Julia set less than 2 for quadratic rational map?
Let $f(z)$ be a quadratic rational map with two Siegel disks which can be normalized to be $$f(z)=z\frac{z+e^{2\pi i\alpha}}{e^{2\pi i\beta}z+1}.$$ If one of the ratation numbers $\alpha$ and $\beta$ ...
1
vote
1answer
94 views
Forster's Theorem
I am looking for a detailed account, in English or French, of the main result and its proof from this paper by O. Forster
Zur Theorie der Steinschen Algebren und Moduln, Mathematische Zeitschrift 97, ...
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votes
0answers
30 views
Which matrix-functions are the derivatives of vector-functions? [migrated]
Let $D$ be a bounded convex open set in $\mathbb{R}^n$ and let
$f:D\to \mathbb{R}^{m\times n}$ be a bounded continuous function
(At the moment I am not interested in problems caused by
complicated ...
1
vote
0answers
92 views
Solving a system of rational functions
Given pairwise distinct numbers $c_1, c_2, \dots c_n \in \mathbb{C} \setminus \{0\}$, does the system of equations $$\frac{6}{c_k} + \sum_{i \ne k} \frac{2}{c_k - c_i} = \sum_{i = 1}^n \frac{1}{c_k - ...
2
votes
2answers
151 views
Does the inverse Laplace transform of the square root exist?
Does the inverse Laplace transform, defined by the integral,
\begin{equation}
F(t) = \mathscr L_s^{-1}\left[\sqrt s\right](t) = \int_{c - i\infty}^{c + i\infty} \sqrt s ~e^{-st} ds
\end{equation}
...
1
vote
2answers
129 views
A Functional Equation concerning analytic functions
Let $P$ be a polynomial and suppose $f : \Bbb{C}\longrightarrow \Bbb{C}$ is a non-constant analytic function such for all $z \in \Bbb{C}, f(z) = f(P(z))$. Clearly when $P$ is linear we can find such ...
3
votes
1answer
129 views
The Integral Trick and An Equality in Nakajima's Lecture
In Nekrasov et al's series papers MNS, they calculate such kinds of integral
$$\frac{E_1 E_2}{N(2\pi i)^N(E_1+E_2) }\oint d\phi_1 \wedge d\phi_2\wedge ...\wedge d\phi_N \prod_{i<N} (-\phi_i) ...
1
vote
1answer
62 views
Two different forms of Schwarz-Christoffel-Mapping of unit disk to rectangle. Are they identical?
I found two different equations for the Schwarz-Christoffel-mapping of a unit disk to a rectangle (which are the general form of the SC-mapping, I guess). The first, e.g. from Link, page 20, is
...
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votes
0answers
32 views
Does an analytic function maps a simple connected region into a simple connected region? [migrated]
Suppose $f$ is analytic, say, in $\mathbb{C}$, and suppose $\Omega$ is a bounded simple connected open domain whose boundary we denote as $\Gamma$, then is $f(\Omega)$ also a simple connected domain ...
2
votes
0answers
71 views
topology generated by irreducible componets of $\Gamma$-invariant closed sets
For an analytic space $U$ equipped with an action of a group $\Gamma$,
call a subset $Z\subseteq U$ $\Gamma$-closed iff
it is a closed analytic subset and each of its irreducible components
is an ...
5
votes
1answer
90 views
Function transformation of exponentials
I came across the following function transformation:
$$
\sum_{j=-\infty}^{\infty} e^{(-j^2\cdot t)} = \sqrt{\frac{\pi}{t}} \cdot \sum_{j=-\infty}^{\infty} e^{(-\frac{\pi^2}{t}\cdot j^2)}
$$
where $ j ...
2
votes
0answers
49 views
How can we describe explicitly the “infinitely complex differentiable” complex-valued local martingales?
Let $\mathcal{F}_t$ be a continuous filtration on a probability space, and let $B$ be a standard $\mathbb{C}$-valued $\mathcal{F}_t$-Brownian motion. Let's call a complex-valued process $X$, possibly ...
0
votes
0answers
46 views
Is it obvious that the defining conditions to obtain a particular singularity are well defined on the quotient space?
Let $~f:\mathbb{C}^2 \rightarrow \mathbb{C}$ be a holomorphic function
vanishing at the origin, with
the following properties:
$$ f_{00}, ~f_{10}, ~f_{01}, ~f_{20}, ~f_{11} =0,~~f_{20} \neq 0 ...
0
votes
0answers
140 views
Local System and Gauss-Manin connection
Fix a complex manifold $X$. Then if we have a line bundle $L=\mathcal{O}(D)$ together with Gauss-Manin connection $\nabla: L \rightarrow L \otimes \Omega^{1}_X$, we get the locally constant sheaf $F$ ...
