Numerische Mathematik

, Volume 136, Issue 3, pp 679–702

# Bessel phase functions: calculation and application

• David E. Horsley
Article

## Abstract

The Bessel phase functions are used to represent the Bessel functions as a positive modulus and an oscillating trigonometric term. This decomposition can be used to aid root-finding of certain combinations of Bessel functions. In this article, we give some new properties of the modulus and phase functions and some asymptotic expansions derived from differential equation theory. We find a bound on the error of the first term of this asymptotic expansion and give a simple numerical method for refining this approximation via standard routines for the Bessel functions. We then show an application of the phase functions to the root finding problem for linear and cross-product combinations of Bessel functions. This method improves upon previous methods and allows the roots in ascending order of these functions to be calculated independently. We give some proofs of correctness and global convergence.

## Mathematics Subject Classification

33C10 33F05 65H05

## Notes

### Acknowledgements

I would like to thank Prof. L.K. Forbes for providing some helpful comments on the manuscript, as well as the two anonymous referees useful remarks and insights. I am particularly grateful to the referee who suggested the works of Segura. This work was supported by an Australian Postgraduate Award at the University of Tasmania.

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