Exercise 9.  Show that the following functions are entire (see Definition 3.2).  

9 (b).  .  

Solution 9 (b).

See text and/or instructor's solution manual.

Answer.    and  .  

The partials are continuous everywhere, so  g(z)  is entire.

Solution.  ,    and  

                 ,          so that  

,     ,     ,     .   

The  Cauchy Riemann equations are  

        ,  

        ,

which hold for all  z.

The partials are continuous everywhere, so   

,  

for all  z.  Therefore,     is an entire function.

We are done.   

Remark.  Later, in Section 5.4 we will show that  

(5-34)                    ,
                    and
(5-35)                    .  

Then    will be an analytic function and its derivative is

.  

We are really done.   

Aside.  We can let Mathematica double check our work.

                    The analytic function    maps this orthogonal grid onto an orthogonal grid.

                    Can you find the subtle difference between the above graphs and those in Exercise 9 (a) ?

This solution is complements of the authors.

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