Exercise 2.  Establish the following identities.

2 (i).   .  

Solution 2 (i).

Solution.   Start with    and obtain   .   

This gives  .   Multiplying by    results in   .  

Use the quadratic formula to solve for    and obtain  

                    ,  

where the fractional power is the multivalued square root function.   

Taking the multivalued log of both sides gives the desired result:

                    .

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