Spherical Bessel function of the first kind: Integral transforms (formula 03.21.22.0008)

SphericalBesselJ

$Mathematica Notation$

Bessel-Type Functions

SphericalBesselJ[nu,z]

Integral transforms

Hankel transforms

http://functions.wolfram.com/03.21.22.0008.01

Input Form

HankelTransform[SphericalBesselJ[\[Nu], t], {t, \[Mu]}, z] == UnitStep[1 - z] (((-1)^(\[Mu]/4) Sqrt[Pi/2] Sqrt[z] ((-(-1)^(3/4)) z)^\[Mu] Gamma[(1/4) (3 + 2 \[Mu] + 2 \[Nu])])/ Gamma[(1/4) (3 - 2 \[Mu] + 2 \[Nu])]) Hypergeometric2F1Regularized[ (1/4) (1 + 2 \[Mu] - 2 \[Nu]), (1/4) (3 + 2 \[Mu] + 2 \[Nu]), 1 + \[Mu], z^2] + UnitStep[z - 1] ((Sqrt[Pi/2] z^(-(3/2) - \[Nu]) Gamma[(1/4) (3 + 2 \[Mu] + 2 \[Nu])])/ Gamma[(1/4) (1 + 2 \[Mu] - 2 \[Nu])]) Hypergeometric2F1Regularized[ (1/4) (3 - 2 \[Mu] + 2 \[Nu]), (1/4) (3 + 2 \[Mu] + 2 \[Nu]), 3/2 + \[Nu], 1/z^2] /; z > 0 && z != 1 && Re[\[Mu] + \[Nu]] > -(3/2)

Standard Form

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MathML Form

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<cn type='integer'> 2 </cn> <ci> ν </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <plus /> <ci> ν </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <neq /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <gt /> <apply> <real /> <apply> <plus /> <ci> μ </ci> <ci> ν </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2007-05-02