Schwarz's symmetry principle

Suppose that f is an analytic function which is defined in the upper half-disk {( left bracketing bar z right bracketing bar )^2<1, ℑ[z]>0}. Further suppose that f extends to a continuous function on the real axis, and takes on real values on the real axis. Then f can be extended to an analytic function on the whole disk by the formula f(z^_) = (f(z))^_, and the values for z reflected across the real axis are the reflections of f(z) across the real axis. It is easy to check that the above function is complex differentiable in the interior of the lower half-disk. What is remarkable is that the resulting function must be analytic along the real axis as well, despite no assumptions of differentiability.

analytic continuation | harmonic function