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10 Bessel FunctionsBessel and Hankel Functions

§10.15 Derivatives with Respect to Order

Noninteger Values of ν

10.15.1 J±ν(z)ν=±J±ν(z)ln(12z)(12z)±νk=0(-1)kψ(k+1±ν)Γ(k+1±ν)(14z2)kk!,
10.15.2 Yν(z)ν=cot(νπ)(Jν(z)ν-πYν(z))-csc(νπ)J-ν(z)ν-πJν(z).

Integer Values of ν

10.15.3 Jν(z)ν|ν=n=π2Yn(z)+n!2(12z)nk=0n-1(12z)kJk(z)k!(n-k).

For Jν(z)/ν at ν=-n combine (10.2.4) and (10.15.3).

10.15.4 Yν(z)ν|ν=n =-π2Jn(z)+n!2(12z)nk=0n-1(12z)kYk(z)k!(n-k),
10.15.5 Jν(z)ν|ν=0 =π2Y0(z),Yν(z)ν|ν=0=-π2J0(z).

Half-Integer Values of ν

For the notations Ci and Si see §6.2(ii). When x>0,

10.15.6 Jν(x)ν|ν=12 =2πx(Ci(2x)sinx-Si(2x)cosx),
10.15.7 Jν(x)ν|ν=-12 =2πx(Ci(2x)cosx+Si(2x)sinx),
10.15.8 Yν(x)ν|ν=12 =2πx(Ci(2x)cosx+(Si(2x)-π)sinx),
10.15.9 Yν(x)ν|ν=-12 =-2πx(Ci(2x)sinx-(Si(2x)-π)cosx).

For further results see Brychkov and Geddes (2005) and Landau (1999, 2000).