Separating Singularities of Holomorphic Functions

1998 ◽  
Vol 41 (4) ◽  
pp. 473-477 ◽  
Author(s):  
Jürgen Müller ◽  
Jochen Wengenroth
Keyword(s):  
Approximation Theory ◽  
Classical Result ◽  
Short Proof ◽  
Holomorphic Functions ◽  
Open Mapping ◽  
Complex Approximation ◽  
Basic Tool ◽  
Open Mapping Theorem ◽  
Separation Results

AbstractWe present a short proof for a classical result on separating singularities of holomorphic functions. The proof is based on the open mapping theorem and the fusion lemma of Roth, which is a basic tool in complex approximation theory. The same method yields similar separation results for other classes of functions.

scholarly journals A PROBABILISTIC PROOF OF THE OPEN MAPPING THEOREM FOR ANALYTIC FUNCTIONS

2013 ◽  
Vol 90 (1) ◽  
pp. 74-76
Author(s):  
GREG MARKOWSKY
Keyword(s):  
Brownian Motion ◽  
Conformal Invariance ◽  
Analytic Functions ◽  
Short Proof ◽  
Open Mapping ◽  
Open Mapping Theorem ◽  
Probabilistic Proof

AbstractThe conformal invariance of Brownian motion is used to give a short proof of the Open Mapping theorem for analytic functions.

scholarly journals Escaping Fatou components of transcendental self-maps of the punctured plane

2019 ◽  
pp. 1-25 ◽  
Author(s):  
DAVID MARTÍ-PETE
Keyword(s):  
Approximation Theory ◽  
Entire Functions ◽  
Holomorphic Functions ◽  
Sufficient Condition ◽  
Wandering Domain ◽  
Simply Connected ◽  
Fatou Components ◽  
Wandering Domains

Abstract We study the iteration of transcendental self-maps of $\,\mathbb{C}^*\!:=\mathbb{C}\setminus \{0\}$ , that is, holomorphic functions $f:\mathbb{C}^*\to\mathbb{C}^*$ for which both zero and infinity are essential singularities. We use approximation theory to construct functions in this class with escaping Fatou components, both wandering domains and Baker domains, that accumulate to $\{0,\infty\}$ in any possible way under iteration. We also give the first explicit examples of transcendental self-maps of $\,\mathbb{C}^*$ with Baker domains and with wandering domains. In doing so, we developed a sufficient condition for a function to have a simply connected escaping wandering domain. Finally, we remark that our results also provide new examples of entire functions with escaping Fatou components.

scholarly journals AN OPEN MAPPING THEOREM

2016 ◽  
Vol 94 (1) ◽  
pp. 65-69
Author(s):  
SAAK S. GABRIYELYAN ◽  
SIDNEY A. MORRIS
Keyword(s):  
Compact Group ◽  
Locally Compact Group ◽  
Open Mapping ◽  
Locally Compact ◽  
Surjective Morphism ◽  
Open Mapping Theorem

It is proved that any surjective morphism $f:\mathbb{Z}^{{\it\kappa}}\rightarrow K$ onto a locally compact group $K$ is open for every cardinal ${\it\kappa}$. This answers a question posed by Hofmann and the second author.

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