Exercise 15.  Given an elegant argument that explains why the following functions are harmonic.  

15 (c).   .  

Solution 15 (c).

See text and/or instructor's solution manual.

Answer.   .  

Solution.   Consider the real part of the hyperbolic identity   ,

The function     is analytic and

                        and    .

By Theorem 3.8 in Section 3.3  both    and    are harmonic functions.

Therefore      is a harmonic function

We are done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

      For this exercise it might seem easier to use Laplace's equation and compute  :  

We are really really done.   

Aside.   Suppose we look at a more complicated function.  

However, if   ,  then there is an advantage to notice that  .  

Although Laplace's equation is satisfied:

The partial derivatives are tedious to compute:

This solution is complements of the authors.

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