Numeric mathematics bessel-functions


Hello numeric fans,

does anybody knows a numerical function, which calculates every root of the bessel function first kind. I have only an algorithm for the hp 50g in USER-RPL which is able to found a root with input an estimation of the position for the root.

It would be more usefull, when one can input let's say: 1. until 5. root and/or when it is possible to find the last root.




Interesting that you pose this question. I am writing an article for the online HP Solve newsletter that discusses a mutli-root finder that locates roots in a range of values. The method can locate roots that are also minima or maxima (something that most root-seeking algorithms fail to do).



Hello Namir,

When your article will be published? I can't wait reading it!

It exists an integral representation for first kind functions, the integrand function can be used to find roots of the first derivate, that's a way to find the last root of the bessel function itself.



The functions Bessel function Jn(z) each have an infinite number of real zeros, all of which are simple with the possible exception of z = 0.

For nonnegative n, the kth positive zeros of first kind Bessel functions are denoted jn,k, except that z = 0 is typically counted as the first zero of J'0(z)(Abramowitz and Stegun 1972, p. 370).

The first few roots jn,k of the Bessel functions are given in the following table for small nonnegative integer values of n and k.

k  J0(x)    J1(x)    J2(x)    J3(x)    J4(x)    J5(x)  
1 2.4048 3.8317 5.1356 6.3802 7.5883 8.7715
2 5.5201 7.0156 8.4172 9.7610 11.0647 12.3386
3 8.6537 10.1735 11.6198 13.0152 14.3725 15.7002
4 11.7915 13.3237 14.7960 16.2235 17.6160 18.9801
5 14.9309 16.4706 17.9598 19.4094 20.8269 22.2178

Sources :


Hello C.Ret,

thank you for your private lesson in mathematics. I forgot the important fact you mentioned. I threw some things to disarray. The approximation for great values of z is written with a cos function, which has obviously an infinit number of roots.

But one problem is left, how to find every roots in a given range. I'm waiting for Namir's article, the suspense is killing me.



Yuo are wel come.

I am also waiting on Namir's article.
This lesson is only a copy-paste from InterNet sources about Bessel functions. But it was a good (or needed) refresch for my memories on the Bessel topic too!

I am glad to see that sharing this can also help anyone else ! :-)

As you, I was a bit disapointed that no algorithm or 'approximation function or code' was available nor advertise on the Web.


I´d suggest you check Jean-Marc Baillard´s authoritative pages. He´s got a few programs worth studying, like the Mac Mahon expansion one, see it here:

It is also included in the BESSEL ROM, availabe at TOS - and by extension on the CL of course; where speed is definitely not an issue.


Thanks a lot.

This Mac Mahon expansion expression si exactly the 'code' I was looking for !


Hello Martin,

it looks like a taylor expansion of the asymptotic approximation, is that right?



Balliard's use of the MacMahon approximation seems to work for roots that are high in their sequence number.

Edited: 1 July 2012, 4:44 p.m.


seems to work for roots that are high in their sequence number.

Can you give an example where it doesn´t work?



MacMahon's formulae are very good for large roots.

I've used the first 5 terms of these asymptotic expansions,

but better formulas are given in the "Digital Library

of Mathematical Functions" up to the terms of order 7


For the first roots, one can use a root-finding program

and ascending series or continued fractions.

Unfortunately, I don't know direct formulas in these cases.

Hope this helps,



Hello JM at al.,

thank you for interesting links and further information.

If I understand right, the formulae from Mac Mahon works even for real k parameter values. For integer values is an easy approach possible (and x --> oo).

again thank you and


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