Re: gp: besselj(nu,x) feature?

[Karim BELABAS <Karim.Belabas@math.u-psud.fr>]

On Tue, 3 Sep 2002, Michael Somos wrote:
>      I think there is a mismatch between the help message and the output:
>
> ? besselj(1,x+O(x^3))
> %1 = 1 - 1/8*x^2 + O(x^3)
> ? besselj(2,x+O(x^3))
> %2 = 1 - 1/12*x^2 + O(x^3)
> ? ?besselj
> besselj(nu,x): J-bessel function of index nu and argument x.
>
> ? \v
>           GP/PARI CALCULATOR Version 2.2.4 (development CHANGES-1.491)
>                 i686 running linux (ix86 kernel) 32-bit version
>               (readline v4.2 enabled, extended help not available)
>
> The J_1(x) needs the x/2 factor and J_2(x) needs the x^2/8 factor.
> Perhaps this was not intended?

I think this was intended [though it would have been much better had it been
documented...]. besselj(nu, x) can be computed for non-integral nu, for which
the x^nu power can't be represented in PARI. So for consistency's sake, it
looks better to omit it altogether in the power series representation. cf.
eta(x).

I've documented this (besselj was not even mentioned in the manual).

    Karim.
-- 
Karim Belabas                    Tel: (+33) (0)1 69 15 57 48
Dép. de Mathematiques, Bat. 425  Fax: (+33) (0)1 69 15 60 19
Université Paris-Sud             Email: Karim.Belabas@math.u-psud.fr
F-91405 Orsay (France)           http://www.math.u-psud.fr/~belabas/
--
PARI/GP Home Page: http://www.parigp-home.de/