the Schwarz reflection principle

Suppose a holomorphic function is defined on the upper half of a region, is continuous down to a segment of the real axis, and takes real values on that segment. The Schwarz reflection principle says you get its values on the lower half for free, by mirroring: reflect across the real axis and conjugate. The function extends holomorphically into the reflected region, with no extra work and no choices to make.

The recipe is concrete. If f is holomorphic on the upper region and real on the real-axis edge, define the extended function on the lower region by F(z) = the conjugate of f(z-bar). Plainly: to evaluate at a point below the axis, reflect that point above the axis (that is the conjugate z-bar), apply f there, and conjugate the answer. Because complex conjugation reverses orientation twice — once in the input, once in the output — the patched function turns out to be holomorphic across the seam, and the realness on the axis is exactly what makes the two halves match continuously.

This is the simplest instance of analytic continuation made geometric, and it generalizes: reflection works across circular arcs too (using inversion in the circle instead of conjugation), provided f maps the arc into a line or circle. It is the standard tool for extending Schwarz-Christoffel maps, for studying functions with real boundary values, and for building automorphisms by symmetry. The honest condition to remember: you need genuine continuity up to the segment and real (or arc-valued) boundary data — without those the reflection need not be holomorphic.