Exercise 16.  Show that the function    in Equation (10-22) is analytic on the ray  .  

Solution 16.

        In Section 10.3.1 we looked at the double-valued function  ,   

and considered expressing    as the function    which is defined by the equation   

(10-22)             ,  

where the principal branch of the square root function is used in both factors.  

Furthermore, it was shown that is continuous on the ray  .

        Now use implicit differentiation to determine the derivative of  .   Square both sides of (10-22) and obtain

                    .

Take the derivatives of both sides

                    ,   

then obtain

                    .

For the point    on the ray    we showed that we have

                    ,   

and the derivative is defined to be

                    .

Hence,    is also continuous on the ray  .

Therefore,    is analytic on the ray  .  

This solution is complements of the authors.

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