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7 Error Functions, Dawson’s and Fresnel IntegralsProperties

§7.7 Integral Representations

Contents

§7.7(i) Error Functions and Dawson’s Integral

Integrals of the type -z2R(z)z, where R(z) is an arbitrary rational function, can be written in closed form in terms of the error functions and elementary functions.

7.7.1 erfcz=2π-z20-z2t2t2+1t,
|phz|14π,
7.7.2 w(z)=1π--t2tt-z=2zπ0-t2tt2-z2,
z>0.
7.7.3 0-at2+2ztt=12πa-z2/a+aF(za),
a>0.
7.7.4 0-att+z2t=πaaz2erfc(az),
a>0, z>0.
7.7.5 01-at2t2+1t=π4a(1-(erfa)2),
a>0.
7.7.6 x-(at2+2bt+c)t=12πa(b2-ac)/aerfc(ax+ba),
a>0.
7.7.7 x-a2t2-(b2/t2)t=π4a(2aberfc(ax+(b/x))+-2aberfc(ax-(b/x))),
x>0, |pha|<14π.
7.7.8 0-a2t2-(b2/t2)t=π2a-2ab,
|pha|<14π, |phb|<14π.
7.7.9 0xerftt=xerfx+1π(-x2-1).

§7.7(ii) Auxiliary Functions

7.7.10 f(z) =1π20-πz2t/2t(t2+1)t,
|phz|14π,
7.7.11 g(z) =1π20t-πz2t/2t2+1t,
|phz|14π,
7.7.12 g(z)+f(z)=-πz2/2zπt2/2t.

Mellin–Barnes Integrals

7.7.13 f(z)=(2π)-3/22πc-c+ζ-sΓ(s)Γ(s+12)Γ(s+34)Γ(14-s)s,
7.7.14 g(z)=(2π)-3/22πc-c+ζ-sΓ(s)Γ(s+12)Γ(s+14)Γ(34-s)s.

In (7.7.13) and (7.7.14) the integration paths are straight lines, ζ=116π2z4, and c is a constant such that 0<c<14 in (7.7.13), and 0<c<34 in (7.7.14).

7.7.15 0-atcos(t2)t=π2f(a2π),
a>0,
7.7.16 0-atsin(t2)t=π2g(a2π),
a>0.

§7.7(iii) Compendia

For other integral representations see Erdélyi et al. (1954a, vol. 1, pp. 265–267, 270), Ng and Geller (1969), Oberhettinger (1974, pp. 246–248), and Oberhettinger and Badii (1973, pp. 371–377).