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7 Error Functions, Dawson’s and Fresnel IntegralsProperties

§7.17 Inverse Error Functions

Contents

§7.17(i) Notation

The inverses of the functions x=erfy, x=erfcy, y, are denoted by

7.17.1 y =inverfx,
y =inverfcx,

respectively.

§7.17(ii) Power Series

With t=12πx,

7.17.2 inverfx=t+13t3+730t5+127630t7+,
|x|<1.

For 25S values of the first 200 coefficients see Strecok (1968).

§7.17(iii) Asymptotic Expansion of inverfcx for Small x

As x0

7.17.3 inverfcxu-1/2+a2u3/2+a3u5/2+a4u7/2+,

where

7.17.4 a2 =18v,
a3 =-132(v2+6v-6),
a4 =1384(4v3+27v2+108v-300),
7.17.5 u=-2/ln(πx2ln(1/x)),

and

7.17.6 v=ln(ln(1/x))-2+lnπ.