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7 Error Functions, Dawson’s and Fresnel IntegralsProperties

§7.18 Repeated Integrals of the Complementary Error Function


§7.18(i) Definition

7.18.1 i-1erfc(z) =2πe-z2,
i0erfc(z) =erfcz,

and for n=0,1,2,,

7.18.2 inerfc(z)=zin-1erfc(t)dt=2πz(t-z)nn!e-t2dt.

§7.18(ii) Graphics

See accompanying text
Figure 7.18.1: Repeated integrals of the scaled complementary error function 2nΓ(12n+1)inerfc(x), n=0,1,2,4,8,16. Magnify

§7.18(iii) Properties

7.18.3 ddzinerfc(z)=-in-1erfc(z),
7.18.4 dndzn(ez2erfcz)=(-1)n2nn!ez2inerfc(z),
7.18.5 d2Wdz2+2zdWdz-2nW=0,

where n=1,2,3,, and A, B are arbitrary constants.

7.18.6 inerfc(z)=k=0(-1)kzk2n-kk!Γ(1+12(n-k)).
7.18.7 inerfc(z)=-znin-1erfc(z)+12nin-2erfc(z),

§7.18(iv) Relations to Other Functions

For the notation see §§18.3, 13.2(i), and 12.2.

Hermite Polynomials

7.18.8 (-1)ninerfc(z)+inerfc(-z)=i-n2n-1n!Hn(iz).

Confluent Hypergeometric Functions

7.18.9 inerfc(z)=e-z2(12nΓ(12n+1)M(12n+12,12,z2)-z2n-1Γ(12n+12)M(12n+1,32,z2)),
7.18.10 inerfc(z)=e-z22nπU(12n+12,12,z2).

The confluent hypergeometric function on the right-hand side of (7.18.10) is multivalued and in the sectors 12π<|phz|<π one has to use the analytic continuation formula (13.2.12).

Parabolic Cylinder Functions

Probability Functions

7.18.12 inerfc(z)=12n-1πHhn(2z).

See Jeffreys and Jeffreys (1956, §§23.081–23.09).

§7.18(v) Continued Fraction

7.18.13 inerfc(z)in-1erfc(z)=1/2z+(n+1)/2z+(n+2)/2z+,

See also Cuyt et al. (2008, p. 269).

§7.18(vi) Asymptotic Expansion

7.18.14 inerfc(z)2πe-z2(2z)n+1m=0(-1)m(2m+n)!n!m!(2z)2m,
z, |phz|34π-δ(<34π).