§7.17 Inverse Error Functions

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§7.17(i) Notation

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

7.17.1	$\displaystyle y$	$\displaystyle=\operatorname{inverf}x,$
7.17.1	$\displaystyle y$	$\displaystyle=\operatorname{inverfc}x,$
ⓘ 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 Ch.7

respectively.

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Keywords:: error functions, inverse functions, power-series expansions
Notes:: See Carlitz (1963).
Permalink:: http://dlmf.nist.gov/7.17.ii
Addition (effective with 1.1.7):: The recurrence relation (7.17.2_5) and some descriptive text above it was added.
See also:: Annotations for §7.17 and Ch.7

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

7.17.2		$\operatorname{inverf}x=t+\tfrac{1}{3}t^{3}+\tfrac{7}{30}t^{5}+\tfrac{127}{630}% t^{7}+\cdots=\sum_{m=0}^{\infty}a_{m}t^{2m+1},$
$\|x\|<1$ ,
ⓘ Symbols: $\operatorname{inverf}\NVar{x}$ : inverse error function and $x$ : real variable Referenced by: Erratum (V1.1.7) for Additions Permalink: http://dlmf.nist.gov/7.17.E2 Encodings: TeX, pMML, png Addition (effective with 1.1.7): The equality “ $=\sum_{m=0}^{\infty}a_{m}t^{2m+1}$ ” was added to this equation. See also: Annotations for §7.17(ii), §7.17 and Ch.7

where $a_{0}=1$ and the other coefficients follow from the recursion

7.17.2_5		$a_{m+1}=\frac{1}{2m+3}\sum_{n=0}^{m}\frac{2n+1}{m-n+1}a_{n}a_{m-n},$
$m=0,1,2,\ldots$ .
ⓘ Symbols: $n$ : nonnegative integer Referenced by: §7.17(ii), Erratum (V1.1.7) for Additions Permalink: http://dlmf.nist.gov/7.17.E2_5 Encodings: TeX, pMML, png Addition (effective with 1.1.7): This equation was added. See also: Annotations for §7.17(ii), §7.17 and Ch.7

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

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Keywords:: asymptotic expansions, error functions, inverse functions
Notes:: (7.17.3) follows from Blair et al. (1976), after modifications.
Permalink:: http://dlmf.nist.gov/7.17.iii
Addition (effective with 1.1.7):: A sentence was added at the end of this subsection indicating that one can find an alternative representation of (7.17.3) in Blair et al. (1976).
See also:: Annotations for §7.17 and Ch.7

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), §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	$\displaystyle a_{2}$	$\displaystyle=\tfrac{1}{8}v,$
	$\displaystyle a_{3}$	$\displaystyle=-\tfrac{1}{32}(v^{2}+6v-6),$
	$\displaystyle a_{4}$	$\displaystyle=\tfrac{1}{384}(4v^{3}+27v^{2}+108v-300),$
ⓘ Defines: $a_{i}$ : coefficients (locally) Symbols: $\mathrm{i}$ : 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, $\ln\NVar{z}$ : 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 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, $\ln\NVar{z}$ : 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 Ch.7

For an alternative representation of (7.17.3) see Blair et al. (1976).