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10 Bessel FunctionsKelvin Functions

§10.61 Definitions and Basic Properties

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§10.61(i) Definitions

Throughout §§10.61–§10.71 it is assumed that x0, ν, and n is a nonnegative integer.

10.61.1 berνx+ibeiνx=Jν(xe3πi/4)=eνπiJν(xe-πi/4)=eνπi/2Iν(xeπi/4)=e3νπi/2Iν(xe-3πi/4),
10.61.2 kerνx+ikeiνx=e-νπi/2Kν(xeπi/4)=12πiHν(1)(xe3πi/4)=-12πie-νπiHν(2)(xe-πi/4).

When ν=0 suffices on ber, bei, ker, and kei are usually suppressed.

Most properties of berνx, beiνx, kerνx, and keiνx follow straightforwardly from the above definitions and results given in preceding sections of this chapter.

§10.61(ii) Differential Equations

10.61.3 x2d2wdx2+xdwdx-(ix2+ν2)w=0,
w=berνx+ibeiνx,ber-νx+ibei-νxkerνx+ikeiνx,ker-νx+ikei-νx.
10.61.4 x4d4wdx4+2x3d3wdx3-(1+2ν2)(x2d2wdx2-xdwdx)+(ν4-4ν2+x4)w=0,
w=ber±νx,bei±νx,ker±νx,kei±νx.

§10.61(iii) Reflection Formulas for Arguments

In general, Kelvin functions have a branch point at x=0 and functions with arguments xe±πi are complex. The branch point is absent, however, in the case of berν and beiν when ν is an integer. In particular,

10.61.5 bern(-x) =(-1)nbernx,
bein(-x) =(-1)nbeinx.

§10.61(iv) Reflection Formulas for Orders

10.61.6 ber-νx =cos(νπ)berνx+sin(νπ)beiνx+(2/π)sin(νπ)kerνx,
bei-νx =-sin(νπ)berνx+cos(νπ)beiνx+(2/π)sin(νπ)keiνx.
10.61.7 ker-νx =cos(νπ)kerνx-sin(νπ)keiνx,
kei-νx =sin(νπ)kerνx+cos(νπ)keiνx.
10.61.8 ber-nx =(-1)nbernx, bei-nx
=(-1)nbeinx,
ker-nx =(-1)nkernx, kei-nx
=(-1)nkeinx.

§10.61(v) Orders ±12

10.61.9 ber12(x2) =2-34πx(excos(x+π8)-e-xcos(x-π8)),
bei12(x2) =2-34πx(exsin(x+π8)+e-xsin(x-π8)).
10.61.10 ber-12(x2) =2-34πx(exsin(x+π8)-e-xsin(x-π8)),
bei-12(x2) =-2-34πx(excos(x+π8)+e-xcos(x-π8)).
10.61.11 ker12(x2) =kei-12(x2)
=-2-34πxe-xsin(x-π8),
10.61.12 kei12(x2) =-ker-12(x2)
=-2-34πxe-xcos(x-π8).