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

抄録

The aim of this paper is to show that the methods discussed in Part I to estimate the zeros of linear homogeneous forms with respect to Hankel function and its derivative is applicable also to the case of higher than linear forms. As a typical form we take the bilinear form
f₂(ν)≡\varphi₁(ν²,a,m)H_ν⁽²⁾(a)H_ν⁽²⁾(ma)+\varphi₂(ν²,a,m)dH_ν⁽²⁾(a)⁄da·H_ν⁽²⁾(ma)
+\varphi₃(ν²,a,m)H_ν⁽²⁾(a)dH_ν⁽²⁾(ma)⁄d(ma)
+\varphi₄(ν²,a,m)dH_ν⁽²⁾(a)⁄da·dH_ν⁽²⁾(ma)⁄d(ma)
qua function of ν. As an example we discuss in detail the form such that
\varphi₁≡1−\frac4ν²a²+\frac4ν²(ν²−9⁄4)a⁴,\varphi₂≡\frac2a\left(1+\fracν²−9⁄4a²\ ight),
\varphi₃≡\frac4ma\left(1−\fracν²−9⁄42a²\ ight),\varphi₄≡−\frac4ma²\left(ν²−\frac94\ ight)
with m<1⁄\sqrt2, which appears actually in the case of the diffraction of elastic waves by a sphere. This is shown to have as zeros, besides the one whose real part is somewhat greater than a and imaginary part =−O(ae^−a), those such as
ν=a\bigg[1+\left(\frac3ρa\ ight)^2⁄3exp\left(\frac53πi\ ight)+\frac43ρ\sqrt1−m²\frac3ρa+\frac1120^4⁄3\left(\frac3ρa\ ight)exp\left(\frac43πi\ ight)
−\left{−\frac176+\frac12\frac11−m²+\frac163(1−m²)\ ight}\frac43ρ\sqrt1−m²\left(\frac3ρa\ ight)^5⁄3exp\left(\frac23πi\ ight)
+\left{\frac12800+\frac17170ρ²−\frac809ρ₂(1−m²)−\frac29ρ²\frac11−m²+\frac1289ρ²(1−m²)²\ ight}\left(\frac3ρa\ ight)²+···\bigg]
and ν=ma\bigg[1+\frac12\left(\frac3ρma\ ight)^2⁄3exp\left(\frac53πi\ ight)−\frac4m³3ρ\frac(1−m²)^1⁄2(2m²−1)²\left(\frac3ρma\ ight)exp\left(\fracπ2i\ ight)
+\frac1120\left(\frac3ρma\ ight)^4⁄3exp\left(\frac43πi\ ight)−\bigg{\frac16−\frac2m²+12m²−1−\frac12\fracm²1−m²
−\frac16m²3\frac(1−m²)(2m²−1)⁴\bigg}\frac4m³3ρ\frac(1−m²)^1⁄2(2m²−1)²\left(\frac3ρma\ ight)^5⁄3exp\left(\fracπ6i\ ight)
+\bigg{\frac12800−\frac1630ρ²−\bigg(\frac12−2·\frac2m²+12m²−1−\fracm²1−m²−\frac1m²+\frac18m⁴
+\frac12m⁴·\frac(<