Theorem 3.7.   Let   [Graphics:Images/CauchyRiemannMod_gr_1371.gif]   be an analytic function on the domain  [Graphics:Images/CauchyRiemannMod_gr_1372.gif].   

If   [Graphics:Images/CauchyRiemannMod_gr_1373.gif]   for all   [Graphics:Images/CauchyRiemannMod_gr_1374.gif],   then   [Graphics:Images/CauchyRiemannMod_gr_1375.gif]   is constant on  [Graphics:Images/CauchyRiemannMod_gr_1376.gif].  

 

Proof.

 

Proof of Theorem 3.7 is in the book.

 

Complex Analysis for Mathematics and Engineering