Bessel function J₃/₂(x)

Bessel's function: J_p(x) = Σ (−1)^k / (k! Γ(k + p + 1)) (x/2)^(2k+p), and the recurrence relation is

J_(p−1)(x) + J_(p+1)(x) = (2p/x) J_p(x).

J_(1/2) and J_(−1/2)

With Γ(k + 3/2) = (2k + 1)!√π / (2^(2k+1) k!):

J_(1/2)(x) = Σ (−1)^k /(k! Γ(k+3/2)) (x/2)^(2k+1/2)
           = √(2/(πx)) Σ (−1)^k x^(2k+1)/(2k+1)!
           = √(2/(πx)) sin x

With Γ(k + 1/2) = (2k)!√π / (4^k k!):

J_(−1/2)(x) = √(2/(πx)) Σ (−1)^k x^(2k)/(2k)! = √(2/(πx)) cos x

J_(3/2)

Put p = 1/2 in the recurrence:

J_(−1/2) + J_(3/2) = (1/x) J_(1/2)

J_(3/2) = (1/x) J_(1/2) − J_(−1/2)
        = √(2/(πx)) [ (sin x)/x − cos x ]

Hence J_(3/2)(x) = √(2/(πx)) ( sin x / x − cos x ).