Theorem 3.3 (Cauchy-Riemann Equations).  Suppose that  

              

is differentiable at the point .  Then the partial derivatives of  u  and  v  exist at the point , and  

(3-14)            ,   and also  

(3-15)            .  

Equating the real and imaginary parts of Equations (3-14) and (3-15) gives  

(3-16)         and .  

Proof.

Proof of Theorem 3.3 is in the book.

Complex Analysis for Mathematics and Engineering