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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.

The trajectories followed in the complex plane by all the zeros of the Hankel function and those of its derivative, when the order varies continuously along real values, are discussed.

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.