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     ,    

hold, 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