Extensions of Harmonic Functions of the Complex Plane Slit Along a Line Segment.

Let I be a line segment in the complex plane C . We describe a method of constructing a bi-Lipschitz sense-preserving mapping of C onto itself, which is harmonic in C \ I and coincides with a given sufficiently regular function f : I → C . As a result we show that a quasiconformal self-mapping of C which is harmonic in C \ I does not have to be harmonic in C . [ABSTRACT FROM AUTHOR]