Theorem 3.4 (Cauchy-Riemann conditions for differentiability).  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 Cauchy-Riemann equations and hold at , then  f(z)  is differentiable at   and the derivative can be computed with either formula (3-14) or (3-15),  i.e.

            ,  or  

            .  

Proof.

Proof of Theorem 3.4 is in the book.

Complex Analysis for Mathematics and Engineering