Exercise 2.  Establish the following identities.

2 (a).   .  

Solution 2 (a).

Solution.   Start with  ,

so that   ,   or   .   

Solving for    gives   ,

where the fractional power is the multivalued square root function.   

Taking the multivalued log of both sides gives  ,

Hence,  .

Therefore,   .  

Aside.   In calculating the principal value of   ,  Mathematica manipulates this formula as follows:   

                       

Therefore,  .  

We are done.   

Aside.  The principal value of    can be calculated with Mathematica.

This solution is complements of the authors.

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