# §7.17 Inverse Error Functions

## §7.17(i) Notation

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

 7.17.1 $\displaystyle y$ $\displaystyle=\mathop{\mathrm{inverf}\/}\nolimits x,$ $\displaystyle y$ $\displaystyle=\mathop{\mathrm{inverfc}\/}\nolimits x,$ Defines: $\mathop{\mathrm{inverfc}\/}\nolimits\NVar{x}$: inverse complementary error function and $\mathop{\mathrm{inverf}\/}\nolimits\NVar{x}$: 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)

respectively.

## §7.17(ii) Power Series

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

 7.17.2 $\mathop{\mathrm{inverf}\/}\nolimits x=t+\tfrac{1}{3}t^{3}+\tfrac{7}{30}t^{5}+% \tfrac{127}{630}t^{7}+\cdots,$ $|x|<1$. Symbols: $\mathop{\mathrm{inverf}\/}\nolimits\NVar{x}$: 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)

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

## §7.17(iii) Asymptotic Expansion of $\mathop{\mathrm{inverfc}\/}\nolimits x$ for Small $x$

As $x\to 0$

 7.17.3 $\mathop{\mathrm{inverfc}\/}\nolimits x\sim u^{-1/2}+a_{2}u^{3/2}+a_{3}u^{5/2}+% a_{4}u^{7/2}+\cdots,$

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)
 7.17.5 $u=-2/\mathop{\ln\/}\nolimits\!\left(\pi x^{2}\mathop{\ln\/}\nolimits\!\left(1/% x\right)\right),$ Defines: $u$: expansion variable (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\ln\/}\nolimits\NVar{z}$: 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)

and

 7.17.6 $v=\mathop{\ln\/}\nolimits\!\left(\mathop{\ln\/}\nolimits\!\left(1/x\right)% \right)-2+\mathop{\ln\/}\nolimits\pi.$ Defines: $v$: expansion variable (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\ln\/}\nolimits\NVar{z}$: 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)