Extra Example 3.  Find an analytic function  ,  given the harmonic function  .

Extra Solution 3.

We use the Milne-Thomson method to construct the harmonic function  .

The function    is analytic and can be constructed as follows:

                       

Now expand the quantity    and obtain  

                        

Therefore  ,   or  

                  .

We are done.   

          The level curves    and  .  

          The orthogonal grid formed with    and  .    

We are really done.   

                    In Section 11.4 we will prove that the image of an orthogonal grid under an analytic function is an orthogonal grid.  

          It is best to worry about these concepts when we get there because this example involves the inverse transformation  .

          The orthogonal grid formed by the composite image of four rectangular grids under the multivalued inverse function  .  

          The spacing between curves is not the same as in the previous figures because lines in the domain grid are equally spaced.

          A portion of the above grid where the mapping is  ,  and the principal value of   is used (see Section 2.2 for details).  

          Remark.  In Section 2.2 we introduced formulas for powers of z and the mulitvalued function  .  

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