Exercise 17.  Use the fact that    is analytic to show that    is a harmonic function.  

Solution 17.

See text and/or instructor's solution manual.

Answer.   Show that     is the imaginary part of  ,  and therefore harmonic by Theorem 3.8 in Section 3.3.  

Solution.   

                        

Theorem 3.8 in Section 3.3  states that if   is analytic then both    and    are harmonic functions.  

Therefore    is a harmonic function.  

We are done.   

Aside.  We can let Mathematica double check our work.

Aside.  It is always possible to see if Laplace's equation holds.

This solution is complements of the authors.

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