__An Integral of a Product of four Bessel Functions__

Frédéric Chyzak

(Version of January 8, 1998)

In (Glasser, M. L. and Montaldi E. (1994): Some Integrals Involving Bessel Functions,
*J. Math. Anal. Appl.*
,
**183**
:577-590), Glasser and Montaldi compute a closed form for an integral of a product of two Bessel functions, and suggest that their treatment should extend to the following example

,

which is of interest because it contains each of the four types of Bessel functions. This integral is one of numerous integrals containing four (or more) Bessel functions. See for instance (Prudnikov, A. P., Brychkov, Yu. A. and Marichev, O. I. (1986):
*Integrals and Series. Volume 2: Special functions*
, Gordon and Breach; Sec. 2.16.47).

In this session, we deal with the integral above and derive a closed form for it using our
__Mgfun__
package in an intimate interaction with the
__gfun__
package.

`> `
**with(Mgfun);**

`> `
**with(gfun);**

More specifically, the
*gfun*
package will be used to prepare a system of PDE's for the application of
*Mgfun*
functions, and to solve the ODE that is output by the
*Mgfun*
package.

**Search for a System of PDE's Satisfied by the Integrand **

**Integration**

**Resolution of the Final ODE**