§7.17 Inverse Error Functions

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

7.17.1		$y$	$=\mathrm{inverf}x,$
		$y$	$=\mathrm{inverfc}x,$
ⓘ Defines: $\mathrm{inverfc}x$: inverse complementary error function and $\mathrm{inverf}x$: 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 Ch.7

respectively.

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

7.17.2		$$\mathrm{inverf}x=t+\frac{1}{3}{t}^3+\frac{7}{30}{t}^5+\frac{127}{630}{t}^7+\mathrm{\cdots },$$
.
ⓘ Symbols: $\mathrm{inverf}x$: 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 Ch.7

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

As $x\to 0$

7.17.3		$$\mathrm{inverfc}x\sim u^{-1/2}+a_2{u}^{3/2}+a_3{u}^{5/2}+a_4{u}^{7/2}+\mathrm{\cdots },$$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\mathrm{inverfc}x$: inverse complementary error function, $x$: real variable, $a_{\mathrm{i}}$: 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 Ch.7

where

7.17.4		$a_2$	$=\frac{1}{8}v,$
		$a_3$	$=-\frac{1}{32}(v^2+6v-6),$
		$a_4$	$=\frac{1}{384}(4v^3+27v^2+108v-300),$
ⓘ Defines: $a_{\mathrm{i}}$: coefficients (locally) Symbols: $\mathrm{i}$: imaginary unit and $v$: 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 Ch.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, $\ln z$: 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 Ch.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, $\ln z$: 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 Ch.7