Theorem  (Complex form of the Cauchy-Riemann Equations).  Suppose the formula for    involves  .  

We can view    as a function of    and write:  

            .  

The complex form of the Cauchy-Riemann equations is  .

Proof.

Recall the identities    and    that were used in Section 2.1.  Use them and get

Therefore, the complex form of the Cauchy-Riemann equations is  .

The details for this solution are left for the reader to work through in Exercise 16.

This solution is complements of the authors.

This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

(c) 2011 John H. Mathews, Russell W. Howell