Digital Library of Mathematical Functions
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10 Bessel FunctionsKelvin Functions

§10.61 Definitions and Basic Properties

Contents

§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+beiνx=Jν(x3π/4)=νπJν(x-π/4)=νπ/2Iν(xπ/4)=3νπ/2Iν(x-3π/4),
10.61.2 kerνx+keiνx=-νπ/2Kν(xπ/4)=12πHν(1)(x3π/4)=-12π-νπHν(2)(x-π/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 x22wx2+xwx-(x2+ν2)w=0,
w=berνx+beiνx,ber-νx+bei-νxkerνx+keiνx,ker-νx+kei-νx.
10.61.4 x44wx4+2x33wx3-(1+2ν2)(x22wx2-xwx)+(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 x±π 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(xcos(x+π8)--xcos(x-π8)),
bei12(x2) =2-34πx(xsin(x+π8)+-xsin(x-π8)).
10.61.10 ber-12(x2) =2-34πx(xsin(x+π8)--xsin(x-π8)),
bei-12(x2) =-2-34πx(xcos(x+π8)+-xcos(x-π8)).
10.61.11 ker12(x2) =kei-12(x2)
=-2-34πx-xsin(x-π8),
10.61.12 kei12(x2) =-ker-12(x2)
=-2-34πx-xcos(x-π8).