Extra Example 2.  Find an analytic function  ,  given the harmonic function  .

Extra Solution 2.

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 image of three rectangular grids under the multivalued inverse function  ,  

          the three inverse mappings    are used, where  .  

                    The mapping    where

                    The mapping    where

                    The mapping    where

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

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