Example 3.7.  At the beginning of this section (Equation (3-13)) we defined the function  

            [Graphics:Images/CauchyRiemannMod_gr_166.gif].   

Show that this function is differentiable for all z, and find its derivative.

Explore Solution 3.7.

Enter the function f[z] and determine u[x,y] and v[x,y].

[Graphics:../Images/CauchyRiemannMod_gr_174.gif]




[Graphics:../Images/CauchyRiemannMod_gr_175.gif]

 

 

Verify that the Cauchy-Riemann equations hold.

[Graphics:../Images/CauchyRiemannMod_gr_176.gif]




[Graphics:../Images/CauchyRiemannMod_gr_177.gif]

 

 

We see that the Cauchy-Riemann equations hold.

Verify that  [Graphics:../Images/CauchyRiemannMod_gr_178.gif].  

[Graphics:../Images/CauchyRiemannMod_gr_179.gif]




[Graphics:../Images/CauchyRiemannMod_gr_180.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2006 John H. Mathews, Russell W. Howell