Example 3.10.  Show that, if f is is the principal square root function given by  

               

where the domain is restricted to be  ,  then the derivative is given by  

              

for every point in the domain  .  

Explore Solution 3.10.

Enter   and use them to form f[z].  Then verify that polar form of the Cauchy-Riemann equations hold.

  Second, find the derivative using polar formulae    and  .  

We have shown that if    where the domain is restricted to be  ,  then the derivative is given by    where  .  

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