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10 Bessel FunctionsModified Bessel Functions

§10.38 Derivatives with Respect to Order

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

Integer Values of ν

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

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=0n-1(12z)kKk(z)k!(n-k).
10.38.5 Iν(z)ν|ν=0 =-K0(z),
Kν(z)ν|ν=0 =0.

Half-Integer Values of ν

For the notations E1 and Ei see §6.2(i). When x>0,

10.38.6 Iν(x)ν|ν=±12=-12πx(E1(2x)ex±Ei(2x)e-x),
10.38.7 Kν(x)ν|ν=±12=±π2xE1(2x)ex.

For further results see Brychkov and Geddes (2005).