Bessel function J₋₅/₂(x)

Bessel's function of order p: J_p(x) = Σ (−1)^k / (k! Γ(k + p + 1)) · (x/2)^(2k+p). We use the recurrence relation

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

Step 1: J_(1/2) and J_(−1/2)

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

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

Using Γ(k + 1/2) = (2k)!√π / (4^k k!):

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

Step 2: J_(−3/2)

Put p = −1/2 in the recurrence:

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

Step 3: J_(−5/2)

Put p = −3/2:

J_(−5/2) = (−3/x) J_(−3/2) − J_(−1/2)
         = √(2/(πx)) [ (3/x)((cos x)/x + sin x) − cos x ]
         = √(2/(πx)) [ (3 sin x)/x + (3/x² − 1) cos x ]

Since 3/x² − 1 = (3 − x²)/x²,

J_(−5/2)(x) = √(2/(πx)) [ (3/x) sin x + ((3 − x²)/x²) cos x ], as required.

(Checked numerically at x = 1.3: both sides equal 1.7012.)