Exercise 1.  Consider the function   ,   where   .  

1 (a).  Show that, if  ,  the sequence    formed by successive iterations of    via    lies entirely within the upper half-plane.

Solution 1(a).

See text and/or instructor's solution manual.

If   ,   then   

                       

Hence, if   ,   then   ,   

then it follows that   .

        Assume   ,   for some  ,    

then   .  

        Therefore, by mathematical induction,

if  ,  then the sequence formed by successive iterations of via   lies entirely within the upper half-plane.

We are done.   

Remark.  We can use Mathematica to explore this situation.

We are really done.   

Remark.  We can use Mathematica to explore this situation with a computation.

This solution is complements of the authors.

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