Schwarz reflection principle

A theorem that extends a holomorphic function across a boundary by reflecting its values when the function satisfies suitable boundary conditions.

Holomorphic function: A complex-valued function that is complex differentiable at every point of an open domain. The principle extends holomorphicity from one side of a boundary to the other.
Complex plane: The plane whose points represent complex numbers, with real and imaginary coordinates. Reflection is geometric across a line or circle in this plane.
Riemann mapping theorem: A theorem stating that every nonempty simply connected proper open subset of the complex plane is conformally equivalent to the unit disk. Reflection can extend conformal maps across sufficiently regular boundary arcs.
Schwarz reflection across the real axis: The extension defined by f(z) = conjugate of f(conjugate z) below the real axis when f is holomorphic above it and real on the axis. This is the standard case from which the general principle is recognized.
Complex conjugation: The map sending a complex number to its conjugate, reflecting the complex plane across the real axis. The standard reflection formula pairs values at conjugate points.
Open set: A subset of a topological space in which every point has a neighborhood contained in the subset. Holomorphicity is defined on open domains, including both sides of the reflecting boundary.
Conformal map: A holomorphic map with nonzero derivative that preserves angles locally. Extending a conformal map can reveal its behavior beyond the original domain.
Schwarz reflection across a circle: A form of Schwarz reflection that extends holomorphic functions across a circle using inversion in that circle. A Möbius transformation reduces circular reflection to reflection across a line.
Identity theorem: A theorem stating that holomorphic functions agreeing on a set with an accumulation point in their domain agree everywhere on a connected domain. Uniqueness of the extension follows from agreement on the original domain.
Continuous extension: An extension of a function to additional points that preserves continuity there. Common forms of the principle require continuous boundary values before reflection.

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