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10 Bessel FunctionsModified Bessel Functions

§10.38 Derivatives with Respect to Order

10.38.1 I±ν(z)ν=±I±ν(z)ln(12z)(12z)±νk=0ψ(k+1±ν)Γ(k+1±ν)(14z2)kk!,
10.38.2 Kν(z)ν=12πcsc(νπ)(Iν(z)νIν(z)ν)πcot(νπ)Kν(z),
ν.

Integer Values of ν

10.38.3 (1)nIν(z)ν|ν=n=Kn(z)+n!2(12z)nk=0n1(1)k(12z)kIk(z)k!(nk),

For Iν(z)/ν at ν=n combine (10.38.1), (10.38.2), and (10.38.4).

10.38.4 Kν(z)ν|ν=n=n!2(12z)nk=0n1(12z)kKk(z)k!(nk).
10.38.5 Iν(z)ν|ν=0 =K0(z),
Kν(z)ν|ν=0 =0.

Half-Integer Values of ν

For further results see Brychkov and Geddes (2005).