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6 Exponential, Logarithmic, Sine, and Cosine IntegralsProperties

§6.12 Asymptotic Expansions


§6.12(i) Exponential and Logarithmic Integrals

6.12.1 E1(z)e-zz(1-1!z+2!z2-3!z3+),
z, |phz|32π-δ(<32π).

When |phz|12π the remainder is bounded in magnitude by the first neglected term, and has the same sign when phz=0. When 12π|phz|<π the remainder term is bounded in magnitude by csc(|phz|) times the first neglected term. For these and other error bounds see Olver (1997b, pp. 109–112) with α=0.

For re-expansions of the remainder term leading to larger sectors of validity, exponential improvement, and a smooth interpretation of the Stokes phenomenon, see §§2.11(ii)2.11(iv), with p=1.

6.12.2 Ei(x)exx(1+1!x+2!x2+3!x3+),

If the expansion is terminated at the nth term, then the remainder term is bounded by 1+χ(n+1) times the next term. For the function χ see §9.7(i).

The asymptotic expansion of li(x) as x is obtainable from (6.2.8) and (6.12.2).

§6.12(ii) Sine and Cosine Integrals

The asymptotic expansions of Si(z) and Ci(z) are given by (6.2.19), (6.2.20), together with

6.12.3 f(z) 1z(1-2!z2+4!z4-6!z6+),
6.12.4 g(z) 1z2(1-3!z2+5!z4-7!z6+),

as z in |phz|π-δ(<π).

The remainder terms are given by

6.12.5 f(z) =1zm=0n-1(-1)m(2m)!z2m+Rn(f)(z),
6.12.6 g(z) =1z2m=0n-1(-1)m(2m+1)!z2m+Rn(g)(z),

where, for n=0,1,2,,

6.12.7 Rn(f)(z) =(-1)n0e-ztt2nt2+1dt,
6.12.8 Rn(g)(z) =(-1)n0e-ztt2n+1t2+1dt.

When |phz|14π, these remainders are bounded in magnitude by the first neglected terms in (6.12.3) and (6.12.4), respectively, and have the same signs as these terms when phz=0. When 14π|phz|<12π the remainders are bounded in magnitude by csc(2|phz|) times the first neglected terms.

For other phase ranges use (6.4.6) and (6.4.7). For exponentially-improved asymptotic expansions, use (6.5.5), (6.5.6), and §6.12(i).