Univalent function: Difference between revisions

Revision as of 14:54, 5 February 2012

Revision as of 03:02, 14 April 2012

Line 2:

== Examples==

Any mapping <math>\phi_a</math> of the open [[unit disc]] to itself, :<math>\phi_a(z) =\frac{z-a}{1 - \bar{a}z},</math> where <math>|a|\le 1,</math> is univalent.

==Basic properties==

One can prove that if <math>G</math> and <math>\Omega</math> are two open [[connected space|connected]] sets in the complex plane, and

Line 18 ⟶ 16:

==Comparison with real functions==

For [[real number|real]] [[analytic function]]s, unlike for complex analytic (that is, holomorphic) functions, these statements fail to hold. For example, consider the function

Line 26 ⟶ 23:

== References==

* John B. Conway. ''Functions of One Complex Variable I''. Springer-Verlag, New York, 1978. ISBN 0-387-90328-3.

* John B. Conway. ''Functions of One Complex Variable II''. Springer-Verlag, New York, 1996. ISBN 0-387-94460-5.

{{planetmathPlanetMath attribution|title=univalent analytic function|id=5633}}

----

{{planetmath|title=univalent analytic function|id=5633}}

[[Category:Analytic functions]]