Theorem  (Complex form of the Cauchy-Riemann Equations).  Suppose the formula for  [Graphics:Images/CauchyRiemannMod_gr_1377.gif]  involves  [Graphics:Images/CauchyRiemannMod_gr_1378.gif].  

We can view  [Graphics:Images/CauchyRiemannMod_gr_1379.gif]  as a function of  [Graphics:Images/CauchyRiemannMod_gr_1380.gif]  and write:  

            [Graphics:Images/CauchyRiemannMod_gr_1381.gif].  

The complex form of the Cauchy-Riemann equations is  [Graphics:Images/CauchyRiemannMod_gr_1382.gif].

 

Proof.

 

Recall the identities  [Graphics:../Images/CauchyRiemannMod_gr_1383.gif]  and  [Graphics:../Images/CauchyRiemannMod_gr_1384.gif]  that were used in Section 2.1.  Use them and get

                    [Graphics:../Images/CauchyRiemannMod_gr_1385.gif]    

Therefore, the complex form of the Cauchy-Riemann equations is  [Graphics:../Images/CauchyRiemannMod_gr_1386.gif].

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