Solution 12 (a).

See text and/or instructor's solution manual.

Solution.  First part, compute the partial derivatives of   :

                    ,   ,   

                    ,   .

Then substitute them into Laplace's equation:

                       

Therefore    is a harmonic function.  

Second part, compute the partial derivatives of   :

                    ,   ,   

                    ,   .

Then substitute them into Laplace's equation:

                       

Therefore    is a harmonic function.  

We are done.   

Aside.  It is easy to verify that   is the harmonic conjugate of  .  Use the polar form of the Cauchy-Riemann equations  

                    ,    and   

                    .  

Hence     is an analytic function for all  .    

A careful inspection shows that this is the polar form of   ,  which is analytic for all  .  

Hence it follows from Theorem 3.8 that      and     are harmonic functions for all  .

We are really done.   

Aside.  We can let Mathematica double check our work.

This solution is complements of the authors.

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