Exercises for Section 3.3.  Harmonic Functions

Exercise 1.  Determine where the following functions are harmonic.

1 (a).    and   .  
Solution 1 (a).

1 (b).    for  .
Solution 1 (b).

Exercise 2.  Does an analytic function    exist for which   ?   Why or why not?
Solution 2.

Exercise 3.  Let a, b and c be real constants.  Determine a relation among the coefficients that will guarantee that the function    is harmonic.  
Solution 3.

Exercise 4.  Let    for  .  Compute the partial derivatives of     and verify that    satisfies Laplace's equation.  
Solution 4.

Exercise 5.  Find an analytic function    for the following expressions.

5 (a).  .
Solution 5 (a).

5 (b).  .  
Solution 5 (b).

5 (c).  .
Solution 5 (c).

5 (d).  .  
Solution 5 (d).

5 (e).  .
Solution 5 (e).

5 (f).  .
Solution 5 (f).

Exercise 6.  Let    and  .  

Show that    are harmonic functions but that their product    is not a harmonic function.
Solution 6.

Exercise 7.  Let    be harmonic on a region D that is symmetric about the line  .  

Show that    is harmonic on D.  

Hint.  Use the chain rule for differentiation of real functions and note that    is really the function ,  where .
Solution 7.

Exercise 8.  Let    be a harmonic conjugate of  .  Show that   is the harmonic conjugate of   .
Solution 8.

Exercise 9.  Let    be a harmonic conjugate of  .  Show that    is a harmonic function.
Solution 9.

Exercise 10.  Suppose that    is a harmonic conjugate of    and that   is the harmonic conjugate of  .

Show that both    must be constant functions.
Solution 10.

Exercise 11.  Let    be analytic on a domain D that does not contain the origin.  

Use the polar form of the Cauchy-Riemann equations    and  .  

Differentiate them first with respect to and then with respect to r.  Use the results to establish the polar form of Laplace's equation:  

                    .  
Solution 11.

Exercise 12.  Use the polar form of Laplace's equation given in Exercise 11 to show that the following functions are harmonic.

12 (a).     and   .  
Solution 12 (a).

12 (b).     and   .  
Solution 12 (b).

Exercise 13.  The function    is used to determine a field known as a dipole.

13 (a).  Express  F(z)  in the form  .  
Solution 13 (a).

13 (b).  Sketch the equipotentials    and the streamlines  .  
Solution 13 (b).

Exercise 14.  Assume that    is analytic on the domain D and that    on  D.  

Consider the families of level curves     and   ,  

which are the equipotentials and streamlines for the fluid flow  .   

Prove that the two families of curves are orthogonal.  

Hint.  Suppose that    is a point common to the two curves    and  .  

Use the gradients of    and    to show that the normals to the curves are perpendicular.
Solution 14.

Exercise 15.  We introduce the logarithmic function in Section 5.2.   For now, let  .    

Here we have      and   .  

Sketch the equipotentials    and the streamlines    for  .  
Solution 15.

Exercise 16.  Discuss and compare the statements "v(x,y) is harmonic" and "v(x,y) is the imaginary part of an analytic function."  
Solution 16.

Exercise 17.  Milne-Thomson Method for constructing a harmonic conjugate.  

(i)  Given the harmonic function    then construct   

                     .  

Show that under the proper conditions,    is a harmonic conjugate of  ,  and

                       is an analytic function.  
Solution 17 (i).

(ii)  Given the harmonic function    then construct   

                     .  

Show that under the proper conditions,    is a harmonic conjugate of  ,  and

                       is an analytic function.  
Solution 17 (ii).

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