Spherical Bessel function of the first kind: Series representations (formula 03.21.06.0038)

SphericalBesselJ

$Mathematica Notation$

Bessel-Type Functions

SphericalBesselJ[nu,z]

Series representations

Generalized power series

Expansions at z==0

For small integer powers of the function

For the second power

http://functions.wolfram.com/03.21.06.0038.01

Input Form

SphericalBesselJ[\[Nu], z]^2 == Subscript[F, Infinity][z . \[Nu]] /; Subscript[F, n][z, \[Nu]] == ((2^(-2 - 2 \[Nu]) Pi z^(2 \[Nu]))/ Gamma[3/2 + \[Nu]]^2) Sum[((-1)^k Pochhammer[\[Nu] + 1, k] z^(2 k))/ (Pochhammer[\[Nu] + 3/2, k] Pochhammer[2 + 2 \[Nu], k] k!), {k, 0, n}] == SphericalBesselJ[\[Nu], z]^2 + (((-1)^n Sqrt[Pi] z^(2 + 2 n + 2 \[Nu]) Gamma[2 + n + \[Nu]])/2) HypergeometricPFQRegularized[{1, 2 + n + \[Nu]}, {2 + n, 5/2 + n + \[Nu], 3 + n + 2 \[Nu]}, -z^2] && Element[n, Integers] && n >= 0

Standard Form

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MathML Form

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Rule Form

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Date Added to functions.wolfram.com (modification date)

2007-05-02