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10 Bessel FunctionsKelvin Functions

§10.67 Asymptotic Expansions for Large Argument

Contents
  1. §10.67(i) berνx,beiνx,kerνx,keiνx, and Derivatives
  2. §10.67(ii) Cross-Products and Sums of Squares in the Case ν=0

§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 ex/2(π2x)12k=0ak(ν)xkcos(x2+(ν2+k4+18)π),
10.67.2 keiνx ex/2(π2x)12k=0ak(ν)xksin(x2+(ν2+k4+18)π).
10.67.3 berνxex/2(2πx)12k=0ak(ν)xkcos(x2+(ν2+3k418)π)1π(sin(2νπ)kerνx+cos(2νπ)keiνx),
10.67.4 beiνxex/2(2πx)12k=0ak(ν)xksin(x2+(ν2+3k418)π)+1π(cos(2νπ)kerνxsin(2νπ)keiνx).
10.67.5 kerνx ex/2(π2x)12k=0bk(ν)xkcos(x2+(ν2+k418)π),
10.67.6 keiνx ex/2(π2x)12k=0bk(ν)xksin(x2+(ν2+k418)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)kerνxsin(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+1641x23325621x3179781921x4+),
10.67.10 berxbeixberxbeix ex22πx(12+181x+96421x2+395121x3+75819221x4+),
10.67.11 berxberx+beixbeix ex22πx(12381x156421x2455121x3+315819221x4+),
10.67.12 (berx)2+(beix)2 ex22πx(13421x+9641x2+7525621x3+247581921x4+).
10.67.13 ker2x+kei2x π2xex2(11421x+1641x2+3325621x3179781921x4+),
10.67.14 kerxkeixkerxkeix π2xex2(12181x+96421x2395121x3+75819221x4+),
10.67.15 kerxkerx+keixkeix π2xex2(12+381x156421x2+455121x3+315819221x4+),
10.67.16 (kerx)2+(keix)2 π2xex2(1+3421x+9641x27525621x3+247581921x4+).