Schwarz's reflection principle

The Schwarz reflection principle (after Hermann Schwarz ) is a statement of function theory about holomorphic functions . It allows, under certain conditions, to continue a holomorphic function holomorphically by mirroring it on the real axis.

statement

It denotes an open subset (in the subspace topology) of the closed upper half-plane . Let be a continuous function that is holomorphic and only takes real values. Then the function is mirrored ${\displaystyle U\subset \mathbb {C} }$ ${\displaystyle \{z\in \mathbb {C} \mid \mathrm {Im} \,z\geq 0\}}$ ${\displaystyle f\colon U\longrightarrow \mathbb {C} }$ ${\displaystyle \{z\in U\mid \mathrm {Im} \,z>0\}}$ ${\displaystyle U\cap \mathbb {R} }$

${\displaystyle {\tilde {f}}(z):=\left\{{\begin{matrix}f(z),&z\in U,\\{\overline {f({\overline {z}})}},&z\in {\overline {U}},\end{matrix}}\right.}$

holomorphic on , where . ${\displaystyle U\cup {\overline {U}}}$ ${\displaystyle {\overline {U}}=\{z\in \mathbb {C} \mid {\overline {z}}\in U\}}$

Proof idea

The statement follows easily from Morera's theorem . For this one only has to show that the integral of vanishes over every closed triangle located in. ${\displaystyle {\tilde {f}}}$ ${\displaystyle U\cup {\overline {U}}}$

For triangles that are entirely in or entirely in , this follows immediately from the assumed holomorphism. For triangles that have a point or a side in common with the real axis, the statement follows with a continuity argument by shifting the triangle a little up or down. ${\displaystyle \{z\in U\mid \mathrm {Im} \,z>0\}}$ ${\displaystyle \{z\in {\overline {U}}\mid \mathrm {Im} \,z<0\}}$

literature

• R. Remmert: Function theory I Springer Verlag, Berlin Heidelberg New York, 1984