Modulus and phase of the reduced logarithmic derivative of the Hankel function

Abstract

The modulus and phase of the reduced logarithmic derivative of the Hankel function \[ z H v ( 1 ) ′ ( z ) / H v ( 1 ) ( z ) zH_v^{(1)}{}’ (z)/H_v^{(1)}(z) \] for complex variable z and real order v, are investigated. Special attention is paid to the location of saddle points and their trajectories as v varies.