Theorem 3.3 (Cauchy-Riemann Equations).  Suppose that  

(3-14)              [Graphics:Images/CauchyRiemannMod_gr_170.gif],  

is differentiable at the point  [Graphics:Images/CauchyRiemannMod_gr_171.gif].   Then the partial derivatives of  [Graphics:Images/CauchyRiemannMod_gr_172.gif]  exist at the point  [Graphics:Images/CauchyRiemannMod_gr_173.gif],  

and can be used to calculate the derivative at [Graphics:Images/CauchyRiemannMod_gr_174.gif].  That is,    

(3-14)              [Graphics:Images/CauchyRiemannMod_gr_175.gif],     

                        and also  

(3-15)              [Graphics:Images/CauchyRiemannMod_gr_176.gif].  

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

(3-16)              [Graphics:Images/CauchyRiemannMod_gr_177.gif]     and     [Graphics:Images/CauchyRiemannMod_gr_178.gif].  

Proof.

Proof of Theorem 3.3 is in the book.

 

Complex Analysis for Mathematics and Engineering