On the Zeros of Certain Transcendental Functions related to Hankel Functions. Part I

The aim of this paper is to determine the zeros of the form {aligned} f₁(ν){≡}φ₁(ν², a){Hν}⁽²⁾(a)+φ₂(ν², a)d{Hν}⁽²⁾(a)/da {aligned} qua function of ν, where H v (2) ( a ) denotes a Hankel function of order ν and ϕ 1 , ϕ 2 are given functions, a being real positive. By studying the analytical behavior of d log H ν (2) ( a )/ d a the regions where the zeros looked for may exist can be determined. If \(|φ₁(ν²,\ a)|{}|φ₂(ν²,\ a)|\), these regions are shown to be a narrow strip, extending from ν= a to -∞ i , and that obtained by the inversion about the origin, where d log H ν (2) ( a )/ d a changes its value infinite times from zero to infinity. The method of estimating these zeros is established by employing the asymptotic expansions of H ν (2) ( a ), and is discussed in detail especially for the zeros near a . As an example the following form is treated, {aligned} d{Hν}⁽²⁾(a)/da+(n/a){Hν}⁽²⁾(a), {aligned} which occurs actually in the case of the diffraction of waves by a sphere. The first several zeros are shown to be given by {aligned} ν{=}a[1+1/2(3ρ/a)2/3exp(5/3π i)+(1/120-{5n-1}{45ρ²})(3ρ/a)4/3exp(4/3π i)+\{1/2800-{5n-2}{450ρ²}+{(5n-1)²}{(45ρ²)²}\}(3ρ/a)²+⋯], {aligned} where ρ denotes one of the following values, {aligned} 0.6853843,\ 3.9027676,\ 7.0549300,\ 10.2006884,\ 13.3445027,\ ⋯. {aligned}