Theorem 3.5  (Polar Form of the Cauchy-Riemann equations).  Let    be a continuous function that is defined in some neighborhood of the point  .  If all the partial derivatives    are continuous at the point and if the polar form of the Cauchy-Riemann equations,

(3-22)               and   ,    

holds, then is differentiable at  , and we can compute the derivative by using either  

(3-23)            ,    or    

(3-24)            .  

Proof.

Proof of Theorem 3.5 is an exercise in the book.

Complex Analysis for Mathematics and Engineering