Modulus and phase of the reduced logarithmic derivative of the cylindrical Bessel function

The modulus and phase of the reduced logarithmic derivative of the cylindrical Bessel function \[ z J ′ v ( z ) / J v ( z ) , z{J\prime _v}(z)/{J_v}(z), \] 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.