Zeros of Hankel Functions and Poles of Scattering Amplitudes

The complex zeros ν_n(z), n = 1, 2, ··· of H_ν⁽¹⁾(z), dH_ν⁽¹⁾(z)/dz and ${\mathrm{dH}}_{\nu }^{\text{(1)}}{\mathrm{(z)/dz+iZH}}_{\nu }^{\text{(1)}}\mathrm{(z)}$ 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 ν_n(z) are obtained for both large and small values of |z| and for large values of n. In addition, for H_ν⁽¹⁾(z) and dH_ν⁽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 ν_n(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.