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7 Error Functions, Dawson’s and Fresnel IntegralsProperties

§7.7 Integral Representations

  1. §7.7(i) Error Functions and Dawson’s Integral
  2. §7.7(ii) Auxiliary Functions
  3. §7.7(iii) Compendia

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

Integrals of the type ez2R(z)dz, 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=2πez20ez2t2t2+1dt,
7.7.2 w(z)=1πiet2dttz=2zπi0et2dtt2z2,
7.7.3 0eat2+2iztdt=12πaez2/a+iaF(za),
7.7.4 0eatt+z2dt=πaeaz2erfc(az),
a>0, z>0.
7.7.5 01eat2t2+1dt=π4ea(1(erfa)2),
7.7.6 xe(at2+2bt+c)dt=12πae(b2ac)/aerfc(ax+ba),
7.7.7 xea2t2(b2/t2)dt=π4a(e2aberfc(ax+(b/x))+e2aberfc(ax(b/x))),
x>0, |pha|<14π.
7.7.8 0ea2t2(b2/t2)dt=π2ae2ab,
|pha|<14π, |phb|<14π.
7.7.9 0xerftdt=xerfx+1π(ex21).

§7.7(ii) Auxiliary Functions

7.7.10 f(z) =1π20eπz2t/2t(t2+1)dt,
7.7.11 g(z) =1π20teπz2t/2t2+1dt,
7.7.12 g(z)+if(z)=eπiz2/2zeπit2/2dt.

Mellin–Barnes Integrals

7.7.13 f(z)=(2π)3/22πicic+iζsΓ(s)Γ(s+12)Γ(s+34)Γ(14s)ds,
7.7.14 g(z)=(2π)3/22πicic+iζsΓ(s)Γ(s+12)Γ(s+14)Γ(34s)ds.

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 0eatcos(t2)dt=π2f(a2π),
7.7.16 0eatsin(t2)dt=π2g(a2π),

§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).