The principal value is analytic. Then, .
The principal value is defined everywhere except at and is discontinuous on the negative real axis. Hence it can't be analytic at these points. The Cauchy-Riemann condition tells us that the function is anaytic everywhere else.
The complex function is composed by two terms...
(1)
As ojones said, the first term is analytic everywhere except in , the second term everywhere except in . The points and are then the two 'brantch points' of the function (1)...
Kind regards
The general question of the analyticity of the function in the whole complex plane with the only exception of has been discussed here...
http://www.mathhelpforum.com/math-he...st-167358.html
Kind regards