Schwarz reflection principle

Sep 2009
173
0
[FONT=&quot]Hi for all[/FONT]
[FONT=&quot]can u help me in solving this,or at least Give me a hint
[/FONT]
[FONT=&quot]Applications to Schwarz reflection principle [/FONT]; Show that there is no non-constant holomorphic function in the unit disc which is real- valued on the unit circle.
Thanks
 

Opalg

MHF Hall of Honor
Aug 2007
4,039
2,789
Leeds, UK
[FONT=&quot]Hi for all[/FONT]
[FONT=&quot]can u help me in solving this,or at least Give me a hint
[/FONT]
[FONT=&quot]Applications to Schwarz reflection principle [/FONT]; 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.
 
Sep 2009
173
0
Thanks alot

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