Schwarz reflection principle

It states that if an analytic function is defined on the upper half-plane, and has well-defined (non-singular) real values on the real axis, then it can be extended to the conjugate function on the lower half-plane.

is a function that satisfies the above requirements, then its extension to the rest of the complex plane is given by the formula,

Suppose that F is a continuous function on the closed upper half plane

Then the extension formula given above is an analytic continuation to the whole complex plane.

In fact Morera's theorem is well adapted to proving such statements.

Contour integrals involving the extension of F clearly split into two, using part of the real axis.

So, given that the principle is rather easy to prove in the special case from Morera's theorem, understanding the proof is enough to generate other results.