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

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- by Robert L. Pexton and Arno D. Steiger PDF
- Math. Comp.
**32**(1978), 511-518 Request permission

## 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 $\sqrt {l(l + 1)} /\alpha$.

## References

- Robert L. Pexton and Arno D. Steiger,
*Roots of two transcendental equations involving spherical Bessel functions*, Math. Comp.**31**(1977), no. 139, 752–753. MR**438662**, DOI 10.1090/S0025-5718-1977-0438662-0
M. ABRAMOWITZ & I. A. STEGUN,

*Handbook of Mathematical Functions*, National Bureau of Standards Applied Mathematics Series No. 55, U. S. Government Printing Office, Washington, D. C., 1965.

## Additional Information

- © Copyright 1978 American Mathematical Society
- Journal: Math. Comp.
**32**(1978), 511-518 - MSC: Primary 65H10
- DOI: https://doi.org/10.1090/S0025-5718-1978-0488704-2
- MathSciNet review: 0488704