Skip to main content Accessibility help
×
Home
Hostname: page-component-768dbb666b-jrcft Total loading time: 0.717 Render date: 2023-02-02T15:08:29.024Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order

Published online by Cambridge University Press:  24 October 2008

F. W. J. Olver
Affiliation:
National Physical LaboratoryTeddington, Middlesex

Extract

In a recent paper (1) I described a method for the numerical evaluation of zeros of the Bessel functions Jn(z) and Yn(z), which was independent of computed values of these functions. The essence of the method was to regard the zeros ρ of the cylinder function

as a function of t and to solve numerically the third-order non-linear differential equation satisfied by ρ(t). It has since been successfully used to compute ten-decimal values of jn, s, yn, s, the sth positive zeros* of Jn(z), Yn(z) respectively, in the ranges n = 10 (1) 20, s = 1(1) 20. During the course of this work it was realized that the least satisfactory feature of the new method was the time taken for the evaluation of the first three or four zeros in comparison with that required for the higher zeros; the direct numerical technique for integrating the differential equation satisfied by ρ(t) becomes unwieldy for the small zeros and a different technique (described in the same paper) must be employed. It was also apparent that no mere refinement of the existing methods would remove this defect and that a new approach was required if it was to be eliminated. The outcome has been the development of the method to which the first part (§§ 2–6) of this paper is devoted.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1951

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Olver, F. W. J.A new method for the evaluation of zeros of Bessel functions and of other solutions of second-order differential equations. Proc. Cambridge Phil. Soc. 46 (1950), 570–80.CrossRefGoogle Scholar
(2)Watson, G. N.Theory of Bessel Functions (Cambridge, 1944).Google Scholar
(3)Ince, E. L.Ordinary Differential Equations (Dover, New York, 1944).Google Scholar
(4)British Association Mathematical Tables, Part-Vol. B, The Airy Integral (Cambridge, 1946).Google Scholar
(5)Meissel, E.Astr. Nach. cxxviii (1891), cols. 435–8.CrossRefGoogle Scholar
(6)Airey, J. R.The numerical calculation of the roots of the Bessel function J n (x) and its first derivate Phil. Mag. 34 (1917), 189–95.CrossRefGoogle Scholar
(7)Jahnke, E. and Emde, , F. Tables of Functions (Dover, New York, 1945), p. 143.Google Scholar
25
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *