Integrating Bessel functions

[Alge Bra]

Hi,

When I integrate J_0(r) J_2(r) r^3 in Maple, I get an answer in terms of the hypergeometric function. But I want an answer in terms of Bessel functions as in the following:

http://www.wolframalpha.com/input/?i=FunctionExpand%5BIntegrate%5BBesselJ%5B0%2C+r%5D+BesselJ%5B2%2C+r%5D+r%5E3%2C+r%5D%5D

The key is "FunctionExpand" that converts the hypergeometric function to more basic functions.

Is it possible to achieve this result in Maple?

Many thanks in advance!

[G. A. Edgar]

In article <b3f2b94b-6803-4b55...@googlegroups.com>,

Maple 16:

> int(BesselJ(0,r)*BesselJ(2,r)*r^3,r);

r^4 (BesselJ(0, r) BesselJ(2, r) + BesselJ(1, r) BesselJ(3, r))/6

--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/

[acer]

The following works in several older versions (not just the lastest Maple 16),

> restart:

> ee:=int(BesselJ(0,r)*BesselJ(2,r)*r^3,r);

1 4
ee := - r (BesselJ(0, r) BesselJ(2, r) + BesselJ(1, r) BesselJ(3, r))
6

> expand(ee);

1 3 1 4 2 4 2 2
- - r BesselJ(0, r) BesselJ(1, r) - - r BesselJ(0, r) + - r BesselJ(1, r)
3 6 3

1 4 2
- - r BesselJ(1, r)
6

> simplify(%,size);

1 2 // 2\ 2
- - r \\-8 + r / BesselJ(1, r) + 2 BesselJ(0, r) BesselJ(1, r) r
6

2 2\
+ r BesselJ(0, r) /


This is all shown as plaintext output, so that I could post it here.

But in Maple's Standard GUI the output can be made to look much terser, with the output appearing as calls to subscripted J_0 and J_1. If the typsetting level is set to extended. That can be set using the GUI's Tools->Options menu, or with the interface(typesetting=extended) command.

For fun, here's a way to get similar terse printing involving apparent calls to J0 and J1, for plaintext output,

> `print/BesselJ`:=proc(a,b)
> if a=0 then 'J0'(b);
> elif a=1 then 'J1'(b);
> else BesselJ(a,b);
> end if;
> end proc:

simplify(expand(ee),size);

1 2 // 2\ 2 2 2\
- - r \\-8 + r / J1(r) + 2 J0(r) J1(r) r + r J0(r) /
6


acer