Theorem 3.3 (Cauchy-Riemann Equations).  Suppose that  

(3-14)              ,  

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

and can be used to calculate the derivative at .  That is,    

(3-14)              ,     

                        and also  

(3-15)              .  

Equating the real and imaginary parts of Equations (3-14) and (3-15) gives the so-called Cauchy-Riemann Equations:

(3-16)                   and     .  

Proof.

Proof of Theorem 3.3 is in the book.

Complex Analysis for Mathematics and Engineering