Theorem 3.5  (Polar Form of the Cauchy-Riemann equations).  Let  [Graphics:Images/CauchyRiemannMod_gr_1031.gif]  

be a continuous function that is defined in some neighborhood of the point  [Graphics:Images/CauchyRiemannMod_gr_1032.gif].   If all the partial derivatives  

[Graphics:Images/CauchyRiemannMod_gr_1033.gif]  are continuous at the point [Graphics:Images/CauchyRiemannMod_gr_1034.gif],  

and if the polar form of the Cauchy-Riemann equations,

(3-22)            [Graphics:Images/CauchyRiemannMod_gr_1035.gif]     and     [Graphics:Images/CauchyRiemannMod_gr_1036.gif],    

hold, then   [Graphics:Images/CauchyRiemannMod_gr_1037.gif]  is differentiable at  [Graphics:Images/CauchyRiemannMod_gr_1038.gif],  and we can compute the derivative  [Graphics:Images/CauchyRiemannMod_gr_1039.gif]  by using either  

(3-23)            [Graphics:Images/CauchyRiemannMod_gr_1040.gif],     or    

(3-24)            [Graphics:Images/CauchyRiemannMod_gr_1041.gif].  

 

Proof.

 

Proof of Theorem 3.5 is an exercise in the book.

 

Complex Analysis for Mathematics and Engineering