# Schwarz reflection principle

• May 26th 2010, 09:47 AM
raed
Schwarz reflection principle
Hi for all
can u help me in solving this,or at least Give me a hint

Applications to Schwarz reflection principle ; Show that there is no non-constant holomorphic function in the unit disc which is real- valued on the unit circle.
Thanks
• May 26th 2010, 10:34 AM
Opalg
Quote:

Originally Posted by raed
Hi for all
can u help me in solving this,or at least Give me a hint

Applications to Schwarz reflection principle ; Show that there is no non-constant holomorphic function in the unit disc which is real- valued on the unit circle.
Thanks

Transfer the holomorphic function f from the unit disc to the upper half-plane by the conformal map $\displaystyle z = \frac{w-i}{w+i}$ (so that f(z) becomes F(w)). The boundary of the disc (the unit circle) gets transformed to the real axis in the w-plane, and you can then use the Schwarz principle to conclude that F is constant and hence so is f.
• May 26th 2010, 11:06 AM
raed
Thanks alot
Quote:

Originally Posted by Opalg
Transfer the holomorphic function f from the unit disc to the upper half-plane by the conformal map $\displaystyle z = \frac{w-i}{w+i}$ (so that f(z) becomes F(w)). The boundary of the disc (the unit circle) gets transformed to the real axis in the w-plane, and you can then use the Schwarz principle to conclude that F is constant and hence so is f.

Thank u very much