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10 Bessel FunctionsKelvin Functions

§10.67 Asymptotic Expansions for Large Argument

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§10.67(i) berνx,beiνx,kerνx,keiνx, and Derivatives

Define ak(ν) and bk(ν) as in §§10.17(i) and 10.17(ii). Then as x with ν fixed,

10.67.1 kerνx e-x/2(π2x)12k=0ak(ν)xkcos(x2+(ν2+k4+18)π),
10.67.2 keiνx -e-x/2(π2x)12k=0ak(ν)xksin(x2+(ν2+k4+18)π).
10.67.3 berνxex/2(2πx)12k=0ak(ν)xkcos(x2+(ν2+3k4-18)π)-1π(sin(2νπ)kerνx+cos(2νπ)keiνx),
10.67.4 beiνxex/2(2πx)12k=0ak(ν)xksin(x2+(ν2+3k4-18)π)+1π(cos(2νπ)kerνx-sin(2νπ)keiνx).
10.67.5 kerνx -e-x/2(π2x)12k=0bk(ν)xkcos(x2+(ν2+k4-18)π),
10.67.6 keiνx e-x/2(π2x)12k=0bk(ν)xksin(x2+(ν2+k4-18)π).
10.67.7 berνxex/2(2πx)12k=0bk(ν)xkcos(x2+(ν2+3k4+18)π)-1π(sin(2νπ)kerνx+cos(2νπ)keiνx),
10.67.8 beiνxex/2(2πx)12k=0bk(ν)xksin(x2+(ν2+3k4+18)π)+1π(cos(2νπ)kerνx-sin(2νπ)keiνx).

The contributions of the terms in kerνx, keiνx, kerνx, and keiνx on the right-hand sides of (10.67.3), (10.67.4), (10.67.7), and (10.67.8) are exponentially small compared with the other terms, and hence can be neglected in the sense of Poincaré asymptotic expansions (§2.1(iii)). However, their inclusion improves numerical accuracy.

§10.67(ii) Cross-Products and Sums of Squares in the Case ν=0

As x

10.67.9 ber2x+bei2x ex22πx(1+1421x+1641x2-3325621x3-179781921x4+),
10.67.10 berxbeix-berxbeix ex22πx(12+181x+96421x2+395121x3+75819221x4+),
10.67.11 berxberx+beixbeix ex22πx(12-381x-156421x2-455121x3+315819221x4+),
10.67.12 (berx)2+(beix)2 ex22πx(1-3421x+9641x2+7525621x3+247581921x4+).
10.67.13 ker2x+kei2x π2xe-x2(1-1421x+1641x2+3325621x3-179781921x4+),
10.67.14 kerxkeix-kerxkeix -π2xe-x2(12-181x+96421x2-395121x3+75819221x4+),
10.67.15 kerxkerx+keixkeix -π2xe-x2(12+381x-156421x2+455121x3+315819221x4+),
10.67.16 (kerx)2+(keix)2 π2xe-x2(1+3421x+9641x2-7525621x3+247581921x4+).