Exercise 12. Show that is analytic for all z by showing that its real and imaginary parts
satisfy the Cauchy-Riemann sufficient conditions for
See text and/or instructor's solution manual.
, and so
, and ,
, and .
The Cauchy Riemann equations are
which hold for all z.
The partials are continuous everywhere, so
for all .
Indeed, we have and , and
We are done.
Aside. We can let Mathematica double check our work.
This solution is complements of the authors.
(c) 2008 John H. Mathews, Russell W. Howell