# §7.17 Inverse Error Functions

## §7.17(i) Notation

The inverses of the functions $x=\operatorname{erf}y$, $x=\operatorname{erfc}y$, $y\in\mathbb{R}$, are denoted by

 7.17.1 $\displaystyle y$ $\displaystyle=\operatorname{inverf}x,$ $\displaystyle y$ $\displaystyle=\operatorname{inverfc}x,$ ⓘ Defines: $\operatorname{inverfc}\NVar{x}$: inverse complementary error function and $\operatorname{inverf}\NVar{x}$: inverse error function Symbols: $x$: real variable Permalink: http://dlmf.nist.gov/7.17.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 7.17(i), 7.17 and 7

respectively.

## §7.17(ii) Power Series

With $t=\frac{1}{2}\sqrt{\pi}x$,

 7.17.2 $\operatorname{inverf}x=t+\tfrac{1}{3}t^{3}+\tfrac{7}{30}t^{5}+\tfrac{127}{630}% t^{7}+\cdots,$ $|x|<1$. ⓘ Symbols: $\operatorname{inverf}\NVar{x}$: inverse error function and $x$: real variable Permalink: http://dlmf.nist.gov/7.17.E2 Encodings: TeX, pMML, png See also: Annotations for 7.17(ii), 7.17 and 7

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

## §7.17(iii) Asymptotic Expansion of $\operatorname{inverfc}x$ for Small $x$

As $x\to 0$

 7.17.3 $\operatorname{inverfc}x\sim u^{-1/2}+a_{2}u^{3/2}+a_{3}u^{5/2}+a_{4}u^{7/2}+\cdots,$ ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\operatorname{inverfc}\NVar{x}$: inverse complementary error function, $x$: real variable, $a_{i}$: coefficients and $u$: expansion variable Referenced by: §7.17(iii) Permalink: http://dlmf.nist.gov/7.17.E3 Encodings: TeX, pMML, png See also: Annotations for 7.17(iii), 7.17 and 7

where

 7.17.4 $\displaystyle a_{2}$ $\displaystyle=\tfrac{1}{8}v,$ $\displaystyle a_{3}$ $\displaystyle=-\tfrac{1}{32}(v^{2}+6v-6),$ $\displaystyle a_{4}$ $\displaystyle=\tfrac{1}{384}(4v^{3}+27v^{2}+108v-300),$ ⓘ Defines: $a_{i}$: coefficients (locally) Symbols: $v$: expansion variable Permalink: http://dlmf.nist.gov/7.17.E4 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for 7.17(iii), 7.17 and 7
 7.17.5 $u=-2/\ln\left(\pi x^{2}\ln\left(1/x\right)\right),$ ⓘ Defines: $u$: expansion variable (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$: principal branch of logarithm function and $x$: real variable Permalink: http://dlmf.nist.gov/7.17.E5 Encodings: TeX, pMML, png See also: Annotations for 7.17(iii), 7.17 and 7

and

 7.17.6 $v=\ln\left(\ln\left(1/x\right)\right)-2+\ln\pi.$ ⓘ Defines: $v$: expansion variable (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$: principal branch of logarithm function and $x$: real variable Permalink: http://dlmf.nist.gov/7.17.E6 Encodings: TeX, pMML, png See also: Annotations for 7.17(iii), 7.17 and 7