The Geometric Theory of Holomorphic Functions

Steven G. Krantz
Additional contact information
Steven G. Krantz: Washington University in St. Louis, Department of Mathematics

Chapter Chapter 6 in Handbook of Complex Variables, 1999, pp 79-88 from Springer

Abstract: Abstract The main objects of study in this chapter are holomorphic functions h: U→ V, with U and V open in ℂ, that are one-to-one and onto. Such a holomorphic function is called a conformal (or biholomorphic) mapping. The fact that h is supposed to be one-to-one implies that h’ is nowhere zero on U [remember that if h’ vanishes to order k ≥ 0 at a point P ∈ U, then h is (k+1)-to-1 in a small neighborhood of P—see §§5.2.1]. As a result, h-1: V →U is also holomorphic—as we discussed in §§5.2.1. A conformal map h: U → V from one open set to another can be used to transfer holomorphic functions on U to V and vice versa: that is, f : V → ℂ is holomorphic if and only if f o h is holomorphic on U; and g : U → ℂ is holomorphic if and only if g o h -1 is holomorphic on V.

Date: 1999
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4612-1588-2_6

Ordering information: This item can be ordered from
http://www.springer.com/9781461215882

DOI: 10.1007/978-1-4612-1588-2_6

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().