general differentiation formula for spherical bessel function
- To: mathgroup at smc.vnet.net
- Subject: [mg116817] general differentiation formula for spherical bessel function
- From: raj <pianoman2008sg at yahoo.com>
- Date: Tue, 1 Mar 2011 05:22:49 -0500 (EST)
D[SphericalBesselJ[L, k Subscript[r, 2]], {k, 1}] //
hi there!,
i am trying to obtain a general formula for differentiating a spherical bessel function of the form SphericalBesselJ[L, kr] m times with respect to k
D[SphericalBesselJ[L, k Subscript[r, 2]], {k, 1}] //
FullSimplify // Apart=
(L SphericalBesselJ[L, k Subscript[r, 2]])/k -
SphericalBesselJ[1 + L, k Subscript[r, 2]] Subscript[r, 2]
D[SphericalBesselJ[L, k Subscript[r, 2]], {k, 2}] //
FullSimplify // Apart =
((-1 + L) L SphericalBesselJ[L, k Subscript[r, 2]])/k^2 + (
2 SphericalBesselJ[1 + L, k Subscript[r, 2]] Subscript[r, 2])/k -
SphericalBesselJ[L, k Subscript[r, 2]] \!
\*SubsuperscriptBox[\(r\), \(2\), \(2\)]
and so on.
is there a general formula in terms of SphericalBesselJ[L, kr]?