Schwarz reflection principle

The Schwarz reflection principle (after Hermann Schwarz ) is a statement of the functional theory of holomorphic functions. It allows, under certain conditions, a holomorphic function by reflection in the real axis holomorphic continue.

Statement

Denote an open subset ( in the subspace topology ) of the closed upper half-plane. Let be a continuous function which is holomorphic on and takes on only real values ​​. Then the mirror function

Holomorphic on where.

Idea of ​​proof

The statement follows easily from the theorem of Morera. You only have to show that the integral of vanishes on every location in the closed triangle.

For triangles that lie completely in or completely in, this follows immediately from the assumed holomorphy. For triangles that have a point or a page with the real axis together, the assertion follows by a continuity argument by moving the triangle slightly up or down.