7 Error Functions, Dawson’s and Fresnel Integrals Properties 7.17 Inverse Error Functions 7.19 Voigt Functions

§7.18 Repeated Integrals of the Complementary Error Function

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Keywords:: error functions, repeated integrals of, repeated integrals of the complementary error function
Referenced by:: §12.7(ii)
Permalink:: http://dlmf.nist.gov/7.18
See also:: Annotations for Ch.7

§7.18(i) Definition
§7.18(ii) Graphics
§7.18(iii) Properties
§7.18(iv) Relations to Other Functions
§7.18(v) Continued Fraction
§7.18(vi) Asymptotic Expansion

§7.18(i) Definition

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Keywords:: definition, repeated integrals of the complementary error function
Permalink:: http://dlmf.nist.gov/7.18.i
See also:: Annotations for §7.18 and Ch.7

7.18.1	$\displaystyle\mathop{\mathrm{i}^{-1}\mathrm{erfc}}\left(z\right)$	$\displaystyle=\frac{2}{\sqrt{\pi}}e^{-z^{2}},$
7.18.1	$\displaystyle\mathop{\mathrm{i}^{0}\mathrm{erfc}}\left(z\right)$	$\displaystyle=\operatorname{erfc}z,$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\mathrm{e}$ : base of natural logarithm, $\mathop{\mathrm{i}^{\NVar{n}}\mathrm{erfc}}\left(\NVar{z}\right)$ : repeated integrals of the complementary error function and $z$ : complex variable A&S Ref: 7.2.1 (in modified form) Referenced by: §7.22(iii) Permalink: http://dlmf.nist.gov/7.18.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §7.18(i), §7.18 and Ch.7

and for $n=0,1,2,\dots$ ,

7.18.2		$\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)=\int_{z}^{\infty}\mathop{% \mathrm{i}^{n-1}\mathrm{erfc}}\left(t\right)\mathrm{d}t=\frac{2}{\sqrt{\pi}}% \int_{z}^{\infty}\frac{(t-z)^{n}}{n!}e^{-t^{2}}\mathrm{d}t.$
ⓘ Defines: $\mathop{\mathrm{i}^{\NVar{n}}\mathrm{erfc}}\left(\NVar{z}\right)$ : repeated integrals of the complementary error function Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\int$ : integral, $z$ : complex variable and $n$ : nonnegative integer A&S Ref: 7.2.1 (in modified form) 7.2.3 Permalink: http://dlmf.nist.gov/7.18.E2 Encodings: TeX, pMML, png See also: Annotations for §7.18(i), §7.18 and Ch.7

§7.18(ii) Graphics

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Keywords:: repeated integrals of the complementary error function, scaled
Notes:: These graphs were produced at NIST.
Permalink:: http://dlmf.nist.gov/7.18.ii
See also:: Annotations for §7.18 and Ch.7

See accompanying text — Figure 7.18.1: Repeated integrals of the scaled complementary error function $2^{n}\Gamma\left(\tfrac{1}{2}n+1\right)\mathop{\mathrm{i}^{n}\mathrm{erfc}}% \left(x\right)$ , $n=0,1,2,4,8,16$ . Magnify

§7.18(iii) Properties

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Keywords:: derivatives, differential equation, power-series expansion, recurrence relations, repeated integrals of the complementary error function
Notes:: See Hartree (1936).
Permalink:: http://dlmf.nist.gov/7.18.iii
See also:: Annotations for §7.18 and Ch.7

7.18.3		$\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z% \right)=-\mathop{\mathrm{i}^{n-1}\mathrm{erfc}}\left(z\right),$
$n=0,1,2,\dots$ ,
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathop{\mathrm{i}^{\NVar{n}}\mathrm{erfc}}\left(\NVar{z}\right)$ : repeated integrals of the complementary error function, $z$ : complex variable and $n$ : nonnegative integer A&S Ref: 7.2.8 Permalink: http://dlmf.nist.gov/7.18.E3 Encodings: TeX, pMML, png See also: Annotations for §7.18(iii), §7.18 and Ch.7

7.18.4		$\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{z^{2}}\operatorname{erfc}z% \right)=(-1)^{n}2^{n}n!e^{z^{2}}\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z% \right),$
$n=0,1,2,\dots$ .
ⓘ Symbols: $\operatorname{erfc}\NVar{z}$ : complementary error function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\mathop{\mathrm{i}^{\NVar{n}}\mathrm{erfc}}\left(\NVar{z}\right)$ : repeated integrals of the complementary error function, $z$ : complex variable and $n$ : nonnegative integer A&S Ref: 7.2.9 Permalink: http://dlmf.nist.gov/7.18.E4 Encodings: TeX, pMML, png See also: Annotations for §7.18(iii), §7.18 and Ch.7

7.18.5		$\frac{{\mathrm{d}}^{2}W}{{\mathrm{d}z}^{2}}+2z\frac{\mathrm{d}W}{\mathrm{d}z}-% 2nW=0,$
$W(z)=A\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)+B\mathop{\mathrm{i}^{% n}\mathrm{erfc}}\left(-z\right)$ ,
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathop{\mathrm{i}^{\NVar{n}}\mathrm{erfc}}\left(\NVar{z}\right)$ : repeated integrals of the complementary error function, $z$ : complex variable and $n$ : nonnegative integer A&S Ref: 7.2.2 Permalink: http://dlmf.nist.gov/7.18.E5 Encodings: TeX, pMML, png See also: Annotations for §7.18(iii), §7.18 and Ch.7

where $n=1,2,3,\dots$ , and $A$ , $B$ are arbitrary constants.

7.18.6		$\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)=\sum_{k=0}^{\infty}\frac{(-% 1)^{k}z^{k}}{2^{n-k}k!\Gamma\left(1+\frac{1}{2}(n-k)\right)}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $!$ : factorial (as in $n!$ ), $\mathop{\mathrm{i}^{\NVar{n}}\mathrm{erfc}}\left(\NVar{z}\right)$ : repeated integrals of the complementary error function, $z$ : complex variable and $n$ : nonnegative integer A&S Ref: 7.2.4 Permalink: http://dlmf.nist.gov/7.18.E6 Encodings: TeX, pMML, png See also: Annotations for §7.18(iii), §7.18 and Ch.7

7.18.7		$\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)=-\frac{z}{n}\mathop{\mathrm% {i}^{n-1}\mathrm{erfc}}\left(z\right)+\frac{1}{2n}\mathop{\mathrm{i}^{n-2}% \mathrm{erfc}}\left(z\right),$
$n=1,2,3,\dots$ .
ⓘ Symbols: $\mathop{\mathrm{i}^{\NVar{n}}\mathrm{erfc}}\left(\NVar{z}\right)$ : repeated integrals of the complementary error function, $z$ : complex variable and $n$ : nonnegative integer A&S Ref: 7.2.5 Referenced by: §7.22(iii) Permalink: http://dlmf.nist.gov/7.18.E7 Encodings: TeX, pMML, png See also: Annotations for §7.18(iii), §7.18 and Ch.7

§7.18(iv) Relations to Other Functions

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Notes:: See Hartree (1936).
Referenced by:: Erratum (V1.0.21) for Paragraph Confluent Hypergeometric Functions (in §7.18(iv))
Permalink:: http://dlmf.nist.gov/7.18.iv
See also:: Annotations for §7.18 and Ch.7

For the notation see §§18.3, 13.2(i), and 12.2.

Hermite Polynomials

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Keywords:: Hermite polynomials, relations to other functions, repeated integrals of the complementary error function
See also:: Annotations for §7.18(iv), §7.18 and Ch.7

7.18.8		$(-1)^{n}\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)+\mathop{\mathrm{i}^% {n}\mathrm{erfc}}\left(-z\right)=\frac{i^{-n}}{2^{n-1}n!}H_{n}\left(iz\right).$
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $\mathop{\mathrm{i}^{\NVar{n}}\mathrm{erfc}}\left(\NVar{z}\right)$ : repeated integrals of the complementary error function, $z$ : complex variable and $n$ : nonnegative integer A&S Ref: 7.2.11 Permalink: http://dlmf.nist.gov/7.18.E8 Encodings: TeX, pMML, png See also: Annotations for §7.18(iv), §7.18(iv), §7.18 and Ch.7

Confluent Hypergeometric Functions

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Keywords:: confluent hypergeometric functions, relations to other functions, repeated integrals of error functions, repeated integrals of the complementary error function
Referenced by:: Erratum (V1.0.21) for Paragraph Confluent Hypergeometric Functions (in §7.18(iv))
Clarification (effective with 1.0.21):: A note about the multivalued nature of (7.18.10) was inserted just below it.
See also:: Annotations for §7.18(iv), §7.18 and Ch.7

7.18.9		$\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)=e^{-z^{2}}\left(\frac{1}{2^% {n}\Gamma\left(\tfrac{1}{2}n+1\right)}M\left(\tfrac{1}{2}n+\tfrac{1}{2},\tfrac% {1}{2},z^{2}\right)-\frac{z}{2^{n-1}\Gamma\left(\tfrac{1}{2}n+\tfrac{1}{2}% \right)}M\left(\tfrac{1}{2}n+1,\tfrac{3}{2},z^{2}\right)\right),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $\mathop{\mathrm{i}^{\NVar{n}}\mathrm{erfc}}\left(\NVar{z}\right)$ : repeated integrals of the complementary error function, $z$ : complex variable and $n$ : nonnegative integer A&S Ref: 7.2.12 Permalink: http://dlmf.nist.gov/7.18.E9 Encodings: TeX, pMML, png See also: Annotations for §7.18(iv), §7.18(iv), §7.18 and Ch.7

7.18.10		$\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)=\frac{e^{-z^{2}}}{2^{n}% \sqrt{\pi}}U\left(\tfrac{1}{2}n+\tfrac{1}{2},\tfrac{1}{2},z^{2}\right).$
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathop{\mathrm{i}^{\NVar{n}}\mathrm{erfc}}\left(\NVar{z}\right)$ : repeated integrals of the complementary error function, $z$ : complex variable and $n$ : nonnegative integer Referenced by: §7.18(iv), §7.18(iv), Erratum (V1.0.21) for Paragraph Confluent Hypergeometric Functions (in §7.18(iv)) Permalink: http://dlmf.nist.gov/7.18.E10 Encodings: TeX, pMML, png See also: Annotations for §7.18(iv), §7.18(iv), §7.18 and Ch.7

The confluent hypergeometric function on the right-hand side of (7.18.10) is multivalued and in the sectors $\tfrac{1}{2}\pi<\left|\operatorname{ph}z\right|<\pi$ one has to use the analytic continuation formula (13.2.12).

Parabolic Cylinder Functions

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Keywords:: parabolic cylinder functions, relations to other functions, repeated integrals of the complementary error function
See also:: Annotations for §7.18(iv), §7.18 and Ch.7

7.18.11		$\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)=\frac{e^{-z^{2}/2}}{\sqrt{2% ^{n-1}\pi}}U\left(n+\tfrac{1}{2},z\sqrt{2}\right).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $\mathop{\mathrm{i}^{\NVar{n}}\mathrm{erfc}}\left(\NVar{z}\right)$ : repeated integrals of the complementary error function, $z$ : complex variable and $n$ : nonnegative integer A&S Ref: 7.2.13 Referenced by: §12.7(ii) Permalink: http://dlmf.nist.gov/7.18.E11 Encodings: TeX, pMML, png See also: Annotations for §7.18(iv), §7.18(iv), §7.18 and Ch.7

Probability Functions

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Keywords:: parabolic cylinder functions, probability functions, relations to other functions, repeated integrals of the complementary error function
See also:: Annotations for §7.18(iv), §7.18 and Ch.7

7.18.12		$\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)=\frac{1}{\sqrt{2^{n-1}\pi}}% \mathit{Hh}_{n}\left(\sqrt{2}z\right).$
ⓘ Defines: $\mathit{Hh}_{\NVar{n}}\left(\NVar{z}\right)$ : probability function Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathop{\mathrm{i}^{\NVar{n}}\mathrm{erfc}}\left(\NVar{z}\right)$ : repeated integrals of the complementary error function, $z$ : complex variable and $n$ : nonnegative integer A&S Ref: 7.2.10 Referenced by: §12.7(ii) Permalink: http://dlmf.nist.gov/7.18.E12 Encodings: TeX, pMML, png See also: Annotations for §7.18(iv), §7.18(iv), §7.18 and Ch.7

See Jeffreys and Jeffreys (1956, §§23.081–23.09).

§7.18(v) Continued Fraction

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Keywords:: continued fractions, repeated integrals of the complementary error function
Notes:: See Hartree (1936) and Lorentzen and Waadeland (1992, p. 577).
Permalink:: http://dlmf.nist.gov/7.18.v
See also:: Annotations for §7.18 and Ch.7

7.18.13		$\frac{\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)}{\mathop{\mathrm{i}^{% n-1}\mathrm{erfc}}\left(z\right)}=\cfrac{1/2}{z+\cfrac{(n+1)/2}{z+\cfrac{(n+2)% /2}{z+}}}\cdots,$
$\Re z>0$ .
ⓘ Symbols: $\Re$ : real part, $\mathop{\mathrm{i}^{\NVar{n}}\mathrm{erfc}}\left(\NVar{z}\right)$ : repeated integrals of the complementary error function, $z$ : complex variable and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/7.18.E13 Encodings: TeX, pMML, png See also: Annotations for §7.18(v), §7.18 and Ch.7

§7.18(vi) Asymptotic Expansion

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Keywords:: asymptotic expansions, repeated integrals of the complementary error function
Notes:: See Hartree (1936).
Permalink:: http://dlmf.nist.gov/7.18.vi
See also:: Annotations for §7.18 and Ch.7

7.18.14		$\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)\sim\frac{2}{\sqrt{\pi}}% \frac{e^{-z^{2}}}{(2z)^{n+1}}\sum_{m=0}^{\infty}\frac{(-1)^{m}(2m+n)!}{n!m!(2z% )^{2m}},$
$z\to\infty$ , $\|\operatorname{ph}z\|\leq\tfrac{3}{4}\pi-\delta(<\tfrac{3}{4}\pi)$ .
ⓘ Symbols: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\operatorname{ph}$ : phase, $\mathop{\mathrm{i}^{\NVar{n}}\mathrm{erfc}}\left(\NVar{z}\right)$ : repeated integrals of the complementary error function, $z$ : complex variable and $n$ : nonnegative integer A&S Ref: 7.2.14 Permalink: http://dlmf.nist.gov/7.18.E14 Encodings: TeX, pMML, png See also: Annotations for §7.18(vi), §7.18 and Ch.7