Zeros of Hankel Functions and Poles of Scattering Amplitudes

Abstract

The complex zeros νn(z), n = 1, 2, ··· of Hν(1)(z), dHν(1)(z)/dz and dHν(1)(z)/dz+iZHν(1)(z) are investigated. These zeros determine the poles in the scattering amplitudes resulting from scattering of various kinds of waves by spheres and cylinders. Formulas for νn(z) are obtained for both large and small values of |z| and for large values of n. In addition, for Hν(1)(z) and dHν(1)(z)/dz, numerical solutions are found for real z in the interval 0.01 ≤ z ≤ 7 and n = 1, 2, 3, 4, 5. The resulting loci of νn(z) in the complex ν plane are presented. These loci are the trajectories of the so-called Regge poles for scattering by spheres and cylinders.