Exercise 7.  Show that the following are harmonic functions in the right half-plane  .  

7 (a).  .  

Solution 7 (a).

See text and/or instructor's solution manual.

Answer.   ,    and    is analytic for  .  

Solution.   

                        


and    is analytic for  .  

By Theorem 3.8 in Section 3.3 both the real and imaginary parts of an analytic function are harmonic functions.

Therefore   is harmonic, and it follows that is harmonic.

We are 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