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10 Bessel FunctionsBessel and Hankel Functions

§10.8 Power Series

For Jν(z) see (10.2.2) and (10.4.1). When ν is not an integer the corresponding expansions for Yν(z), Hν(1)(z), and Hν(2)(z) are obtained by combining (10.2.2) with (10.2.3), (10.4.7), and (10.4.8).

When n=0,1,2,,

10.8.1 Yn(z)=(12z)nπk=0n1(nk1)!k!(14z2)k+2πln(12z)Jn(z)(12z)nπk=0(ψ(k+1)+ψ(n+k+1))(14z2)kk!(n+k)!,

where ψ(x)=Γ(x)/Γ(x)5.2(i)). In particular,

10.8.2 Y0(z)=2π(ln(12z)+γ)J0(z)+2π(14z2(1!)2(1+12)(14z2)2(2!)2+(1+12+13)(14z2)3(3!)2),

where γ is Euler’s constant (§5.2(ii)).

For negative values of n use (10.4.1).

The corresponding results for Hn(1)(z) and Hn(2)(z) are obtained via (10.4.3) with ν=n.

10.8.3 Jν(z)Jμ(z)=(12z)ν+μk=0(ν+μ+k+1)k(14z2)kk!Γ(ν+k+1)Γ(μ+k+1).

Note that (10.8.3) is just a rewriting of (16.12.1).