Roots of two transcendental equations as functions of a continuous real parameter

Abstract:

The roots, $\lambda$ and $\eta$, of the transcendental equations ${j_l}(\alpha \lambda ){y_l}(\lambda ) = {j_l}(\lambda ){y_l}(\alpha \lambda )$ and \[ [x{j_l}(x)]_{x = \alpha \eta }^\prime [x{y_l}(x)]_{x = \eta }^\prime = [x{j_l}(x)]_{x = \eta }^\prime [x{y_l}(x)]_{x = \alpha \eta }^\prime \] where $l = 1,2, \ldots$ are considered as functions of the continuous real parameter $\alpha$. The symbols ${j_l}$ and ${y_l}$ denote the spherical Bessel functions of the first and second kind. The two transcendental equations are invariant under the transformations $\lambda \to - \lambda$ and $\eta \to - \eta$, respectively. Therefore, only positive roots are discussed. All the $\lambda$-roots increase monotonically as $\alpha$ increases in the open interval (0, 1). For each order l, the smallest $\eta$-root decreases monotonically as $\alpha$ increases in (0, 1), tending towards $\sqrt {l(l + 1)}$ as $\alpha$ approaches unity. For $\alpha \in (0,1)$ all the other $\eta$-roots have a minimum value equal to $\sqrt {l(l + 1)} /\alpha$.