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10 Bessel FunctionsSpherical Bessel Functions

§10.50 Wronskians and Cross-Products

10.50.1 𝒲{jn(z),yn(z)} =z-2,
𝒲{hn(1)(z),hn(2)(z)} =-2iz-2.
10.50.2 𝒲{in(1)(z),in(2)(z)} =(-1)n+1z-2,
𝒲{in(1)(z),kn(z)} =𝒲{in(2)(z),kn(z)}
=-12πz-2.
10.50.3 jn+1(z)yn(z)-jn(z)yn+1(z) =z-2,
jn+2(z)yn(z)-jn(z)yn+2(z) =(2n+3)z-3.
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(n-1)/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).