Solution 9.

See text and/or instructor's solution manual.

Solution.  The function    is an analytic function.

Hence    is an analytic function.

By Theorem 3.8  the real and imaginary parts of    are harmonic functions,  i. e.  

                    both    are harmonic functions.  

Therefore    is a harmonic function.

This solution is complements of the authors.

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