Exercise 11.  Determine where the following functions are differentiable and where they are analytic.  Explain!  

11 (c).  .

Solution 11 (c).

See text and/or instructor's solution manual.

Answer.  f(z)  is differentiable at points on the x-axis and y-axis and in quadrants I and III,

and  f(z)  is analytic at points in quadrants I and III.

Solution.  ,  

                 ,          and  

,     ,     ,     .   

The  Cauchy Riemann equations are  

        ,  

        ,

which hold only for the points that satisfy    and  ,  which implies that  .  

Points in quadrant I where    satisfy  ,  and
points in quadrant III where     satisfy  .

Points in quadrant II where    do not satisfy  ,  and
points in quadrant IV where     do not satisfy  .

Clearly the condition    is satisfied at points on the x-axis and y-axis.

Therefore  f(z)  is differentiable at points on the x-axis and y-axis and in quadrants I and III,

and  f(z)  is analytic at points in quadrants I and III.

We are done.   

Aside.  A reason why f(z) is not analytic is given in Exercise 16.  

     Loosely speaking, if f(z) is analytic then there cannot be any occurrence of the variable  .

Recall the identities    and    that were used in Section 2.1.

They can be substituted in ,  and the result is:

At points on the x-axis and y-axis and in quadrants I and III,

               .

At points in quadrants II and IV,

               .

So technically there are "hidden" terms involving    in the latter formula for  f(z),  and this precludes the possibility that  f(z)  can be analytic in n quadrants II and IV.

Indeed, the complex form of the Cauchy-Riemann equations fails to hold in quadrants II and IV because  .  

We are really done.   

Aside.  We can let Mathematica double check our work.

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

                    The function    maps this orthogonal grid onto an orthogonal grid.
                    Remark.  The image region is the conjugate of the one we obtained using   .

                    The image of the region in quadrant I under    is a region that lies in the upper half plane .

                    The image of the region parts in quadrants II and IV under    is the region parts that lie in the upper half plane .

    Now we combine these images to get the following one.

          The image of this orthogonal grid under    is an "overlaping" orthogonal grid.
           Be careful because the function is not analytic on all if the domain set in the z-plane.  Actually some of the region is
           mapped in a two-to-one manner and some of it is mapped in a one-to-one manner.  

This solution is complements of the authors.

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