Concerning the Zeros of Some Functions Related to Bessel Functions

The real zeros of the Riccati-Bessel functions (12πx)12Jv(x), (12πx)12Yv(x), of their derivatives d[(12πx)12Jv(x)]/dx, d[(12πx)12Yv(x)]/dx,and of their cross products 12πx[Jv(x)Yv(Kx) − Yv(x)Jv(Kx)],ddx[(πx2)12Jv(x)]ddx[(πx2)12Yv(Kx)] − ddx[(πx2)12Yv(x)]ddx[(πx2)12Jv(Kx)]are investigated. Expansions analogous to those provided by McMahon and Olver for the zeros of the Bessel functions are obtained for the zeros of the derivatives of the Riccati-Bessel functions. The analysis of Kalähne for the zeros of the cross product of Bessel functions is considerably expanded and analogous results are obtained for the zeros of the cross product of the derivatives of the Riccati-Bessel functions. Included are derivations of the expansions for large zeros at fixed v, of asymptotic expansions for large v at fixed number of the zero, and also asymptotic expansions for the zeros as K→1 and K→∞. Figures illustrating the behavior of the zeros are provided for v = l + ½, where l is an integer. These zeros correspond to the TE and TM electromagnetic normal modes inside a conducting spherical shell and in the region between two concentric conducting shells.