What's New
About the Project
NIST
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 e-z2R(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=2πe-z20e-z2t2t2+1dt,
|phz|14π,
7.7.2 w(z)=1πi-e-t2dtt-z=2zπi0e-t2dtt2-z2,
z>0.
7.7.3 0e-at2+2iztdt=12πae-z2/a+iaF(za),
a>0.
7.7.4 0e-att+z2dt=πaeaz2erfc(az),
a>0, z>0.
7.7.5 01e-at2t2+1dt=π4ea(1-(erfa)2),
a>0.
7.7.6 xe-(at2+2bt+c)dt=12πae(b2-ac)/aerfc(ax+ba),
a>0.
7.7.7 xe-a2t2-(b2/t2)dt=π4a(e2aberfc(ax+(b/x))+e-2aberfc(ax-(b/x))),
x>0, |pha|<14π.
7.7.8 0e-a2t2-(b2/t2)dt=π2ae-2ab,
|pha|<14π, |phb|<14π.
7.7.9 0xerftdt=xerfx+1π(e-x2-1).

§7.7(ii) Auxiliary Functions

7.7.10 f(z) =1π20e-πz2t/2t(t2+1)dt,
|phz|14π,
7.7.11 g(z) =1π20te-πz2t/2t2+1dt,
|phz|14π,
7.7.12 g(z)+if(z)=e-πiz2/2zeπit2/2dt.

Mellin–Barnes Integrals

7.7.13 f(z)=(2π)-3/22πic-ic+iζ-sΓ(s)Γ(s+12)Γ(s+34)Γ(14-s)ds,
7.7.14 g(z)=(2π)-3/22πic-ic+iζ-sΓ(s)Γ(s+12)Γ(s+14)Γ(34-s)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 0e-atcos(t2)dt=π2f(a2π),
a>0,
7.7.16 0e-atsin(t2)dt=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).