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10 Bessel FunctionsSpherical Bessel Functions

§10.50 Wronskians and Cross-Products

10.50.1 𝒲{𝗃n(z),𝗒n(z)} =z2,
𝒲{𝗁n(1)(z),𝗁n(2)(z)} =2iz2.
10.50.2 𝒲{𝗂n(1)(z),𝗂n(2)(z)} =(1)n+1z2,
𝒲{𝗂n(1)(z),𝗄n(z)} =𝒲{𝗂n(2)(z),𝗄n(z)}=12πz2.
10.50.3 𝗃n+1(z)𝗒n(z)𝗃n(z)𝗒n+1(z) =z2,
𝗃n+2(z)𝗒n(z)𝗃n(z)𝗒n+2(z) =(2n+3)z3.
10.50.4 𝗃0(z)𝗃n(z)+𝗒0(z)𝗒n(z)=cos(12nπ)k=0n/2(1)ka2k(n+12)z2k+2+sin(12nπ)k=0(n1)/2(1)ka2k+1(n+12)z2k+3,

where ak(n+12) is given by (10.49.1).

Results corresponding to (10.50.3) and (10.50.4) for 𝗂n(1)(z) and 𝗂n(2)(z) are obtainable via (10.47.12).