Example 3.8.  Show that the function    is differentiable for all and find its derivative.

Explore Solution 3.8.

Enter the function f[z] and determine if the Cauchy-Riemann equations hold.

The Cauchy-Riemann equations hold everywhere, so that   is analytic for all values of  z.

Remark.  We can write this function as  ,  and investigate  .  

We have shown that   is differentiable and hence analytic for all z.

Remark.    is an entire function.

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