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7 Error Functions, Dawson’s and Fresnel Integrals Properties 7.16 Generalized Error Functions 7.18 Repeated Integrals of the Complementary Error Function

§7.17 Inverse Error Functions

Referenced by:: ¶ ‣ §8.12
Permalink:: http://dlmf.nist.gov/7.17

Contents

§7.17(i) Notation
§7.17(ii) Power Series
§7.17(iii) Asymptotic Expansion of $\mathop{\mathrm{inverfc}\/}\nolimits x$ for Small

§7.17(i) Notation

Keywords:: error functions
Permalink:: http://dlmf.nist.gov/7.17.i

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

7.17.1

$y=\mathop{\mathrm{inverf}\/}\nolimits x,$

$y=\mathop{\mathrm{inverfc}\/}\nolimits x,$

Defines:: $\mathop{\mathrm{inverfc}\/}\nolimits x$ : inverse complementary error function and $\mathop{\mathrm{inverf}\/}\nolimits x$ : inverse error function
Symbols:: : real variable
Permalink:: http://dlmf.nist.gov/7.17.E1
Encodings:: TeX, TeX, pMML, pMML, png, png

respectively.

§7.17(ii) Power Series

Notes:: See Carlitz (1963).
Keywords:: error functions
Permalink:: http://dlmf.nist.gov/7.17.ii

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

7.17.2 $\mathop{\mathrm{inverf}\/}\nolimits x=t+\tfrac{1}{3}t^{3}+\tfrac{7}{30}t^{5}+% \tfrac{127}{630}t^{7}+\cdots,$

.

Symbols:: $\mathop{\mathrm{inverf}\/}\nolimits x$ : inverse error function and : real variable
Permalink:: http://dlmf.nist.gov/7.17.E2
Encodings:: TeX, pMML, png

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

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

Notes:: (7.17.3) follows from Blair et al. (1976), after modifications.
Keywords:: error functions
Permalink:: http://dlmf.nist.gov/7.17.iii

As $x\to 0$

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

Symbols:: $\mathop{\mathrm{inverfc}\/}\nolimits x$ : inverse complementary error function, $\sim$ : asymptotic equality, : real variable, $a_{i}$ : coefficients and : expansion variable
Referenced by:: §7.17(iii)
Permalink:: http://dlmf.nist.gov/7.17.E3
Encodings:: TeX, pMML, png

where

7.17.4

$a_{2}=\tfrac{1}{8}v,$

$a_{3}=-\tfrac{1}{32}(v^{2}+6v-6),$

$a_{4}=\tfrac{1}{384}(4v^{3}+27v^{2}+108v-300),$

Symbols:: $a_{i}$ : coefficients and : expansion variable
Permalink:: http://dlmf.nist.gov/7.17.E4
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

7.17.5 $u=-2/\mathop{\ln\/}\nolimits\!\left(\pi x^{2}\mathop{\ln\/}\nolimits\!\left(1/% x\right)\right),$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : real variable and : expansion variable
Permalink:: http://dlmf.nist.gov/7.17.E5
Encodings:: TeX, pMML, png

and

7.17.6 $v=\mathop{\ln\/}\nolimits\!\left(\mathop{\ln\/}\nolimits\!\left(1/x\right)% \right)-2+\mathop{\ln\/}\nolimits\pi.$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : real variable and : expansion variable
Permalink:: http://dlmf.nist.gov/7.17.E6
Encodings:: TeX, pMML, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 7.16 Generalized Error Functions 7.18 Repeated Integrals of the Complementary Error Function