Example 3.4.  We know that   is differentiable and that  .  We also have  

            .  

It is easy to verify that Equations (3-16) are indeed satisfied:

       and   .  

Using Equations (3-14) and (3-15), respectively, to compute   gives

        ,   and   

        ,

as expected.

Explore Solution 3.4.

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

Verify that the Cauchy-Riemann equations hold.

We see that the Cauchy-Riemann equations hold.

Verify that  .  

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