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10 Bessel FunctionsKelvin Functions

§10.67 Asymptotic Expansions for Large Argument

Contents

§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 -x/2(π2x)12k=0ak(ν)xkcos(x2+(ν2+k4+18)π),
10.67.2 keiνx --x/2(π2x)12k=0ak(ν)xksin(x2+(ν2+k4+18)π).
10.67.3 berνxx/2(2πx)12k=0ak(ν)xkcos(x2+(ν2+3k4-18)π)-1π(sin(2νπ)kerνx+cos(2νπ)keiνx),
10.67.4 beiνxx/2(2πx)12k=0ak(ν)xksin(x2+(ν2+3k4-18)π)+1π(cos(2νπ)kerνx-sin(2νπ)keiνx).
10.67.5 kerνx --x/2(π2x)12k=0bk(ν)xkcos(x2+(ν2+k4-18)π),
10.67.6 keiνx -x/2(π2x)12k=0bk(ν)xksin(x2+(ν2+k4-18)π).
10.67.7 berνxx/2(2πx)12k=0bk(ν)xkcos(x2+(ν2+3k4+18)π)-1π(sin(2νπ)kerνx+cos(2νπ)keiνx),
10.67.8 beiνxx/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 x22πx(1+1421x+1641x2-3325621x3-179781921x4+),
10.67.10 berxbeix-berxbeix x22πx(12+181x+96421x2+395121x3+75819221x4+),
10.67.11 berxberx+beixbeix x22πx(12-381x-156421x2-455121x3+315819221x4+),
10.67.12 (berx)2+(beix)2 x22πx(1-3421x+9641x2+7525621x3+247581921x4+).
10.67.13 ker2x+kei2x π2x-x2(1-1421x+1641x2+3325621x3-179781921x4+),
10.67.14 kerxkeix-kerxkeix -π2x-x2(12-181x+96421x2-395121x3+75819221x4+),
10.67.15 kerxkerx+keixkeix -π2x-x2(12+381x-156421x2+455121x3+315819221x4+),
10.67.16 (kerx)2+(keix)2 π2x-x2(1+3421x+9641x2-7525621x3+247581921x4+).