Bessel function of the first kind: Series representations (formula 03.01.06.0052)

BesselJ

$Mathematica Notation$

Bessel-Type Functions

BesselJ[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/03.01.06.0052.01

Input Form

BesselJ[\[Nu], z]^2 == Subscript[F, Infinity][z . \[Nu]] /; Subscript[F, n][z, \[Nu]] == (z^(2 \[Nu])/(2^(2 \[Nu]) Gamma[1 + \[Nu]]^2)) Sum[((-1)^k Pochhammer[1/2 + \[Nu], k] z^(2 k))/(Pochhammer[1 + \[Nu], k] Pochhammer[1 + 2 \[Nu], k] k!), {k, 0, n}] == BesselJ[\[Nu], z]^2 + (((-1)^n z^(2 + 2 n + 2 \[Nu]) Gamma[3/2 + n + \[Nu]])/(Sqrt[Pi] Gamma[2 + n + \[Nu]] Gamma[2 + n + 2 \[Nu]] (n + 1)!)) HypergeometricPFQ[ {1, 3/2 + n + \[Nu]}, {2 + n, 2 + n + \[Nu], 2 + 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