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general formula for differentiating a spherical bessel function of

  • To: mathgroup at smc.vnet.net
  • Subject: [mg122021] general formula for differentiating a spherical bessel function of
  • From: raj kumar <rajesh7796gm at gmail.com>
  • Date: Sun, 9 Oct 2011 03:52:56 -0400 (EDT)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com

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]?



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