Exercise 7.  Find the derivatives of the following and state where they are defined.  

7 (a).   .  

Solution 7 (a).

See text and/or instructor's solution manual.

Answer.   ,   is valid for  .  

Solution.   Use the chain rule and obtain  

                      

We are done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

Aside.  When we apply the "rules for differentiation" we miss the point that we are working with functions of a complex variable.

Would we want to use the "Cauchy-Riemann" equations for finding a derivative of a complex function?  Would it be tractable?

These details are given below with the assistance of Mathematica.

We are grateful that the "ordinary rules of differentiation" extend to analytic functions.

First, expand the given function    in its    form:

Second, find   with the Cauchy-Riemann formula (3-14)    that was given in Section 3.2.

Third, expand the derivative    we found in its    form:

Fourth, observe that the two versions of the derivative are the same.

Remark.  Without the assistance of computer algebra software we might not have given this demonstration.  

We are really really done.   

Aside.  For this exercise it is possible to establish the identity      using series.

Use the known series representations        and       

and the substitution    and obtain:

                    ,    and  


                    .  

Now use termwise differentiation of series and get:

                      

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell