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7 Error Functions, Dawson’s and Fresnel IntegralsProperties

§7.12 Asymptotic Expansions

Contents

§7.12(i) Complementary Error Function

As z

7.12.1 erfcz e-z2πzm=0(-1)m135(2m-1)(2z2)m,
erfc(-z) 2-e-z2πzm=0(-1)m135(2m-1)(2z2)m,

both expansions being valid when |phz|34π-δ (<34π).

When |phz|14π the remainder terms are bounded in magnitude by the first neglected terms, and have the same sign as these terms when phz=0. When 14π|phz|<12π the remainder terms are bounded in magnitude by csc(2|phz|) times the first neglected terms. For these and other error bounds see Olver (1997b, pp. 109–112), with α=12 and z replaced by z2; compare (7.11.2).

For re-expansions of the remainder terms leading to larger sectors of validity, exponential improvement, and a smooth interpretation of the Stokes phenomenon, see §§2.11(ii)2.11(iv) and use (7.11.3). (Note that some of these re-expansions themselves involve the complementary error function.)

§7.12(ii) Fresnel Integrals

The asymptotic expansions of C(z) and S(z) are given by (7.5.3), (7.5.4), and

7.12.2 f(z)1πzm=0(-1)m135(4m-1)(πz2)2m,
7.12.3 g(z)1π2z3m=0(-1)m135(4m+1)(πz2)2m,

as z in |phz|12π-δ(<12π). The remainder terms are given by

7.12.4 f(z)=1πzm=0n-1(-1)m13(4m-1)(πz2)2m+Rn(f)(z),
7.12.5 g(z)=1π2z3m=0n-1(-1)m13(4m+1)(πz2)2m+Rn(g)(z),

where, for n=0,1,2, and |phz|<14π,

7.12.6 Rn(f)(z)=(-1)nπ20e-πz2t/2t2n-(1/2)t2+1dt,
7.12.7 Rn(g)(z)=(-1)nπ20e-πz2t/2t2n+(1/2)t2+1dt.

When |phz|18π, Rn(f)(z) and Rn(g)(z) are bounded in magnitude by the first neglected terms in (7.12.2) and (7.12.3), respectively, and have the same signs as these terms when phz=0. They are bounded by |csc(4phz)| times the first neglected terms when 18π|phz|<14π.

For other phase ranges use (7.4.7) and (7.4.8). For exponentially-improved expansions use (7.5.7), (7.5.10), and §7.12(i).

§7.12(iii) Goodwin–Staton Integral

See Olver (1997b, p. 115) for an expansion of G(z) with bounds for the remainder for real and complex values of z.