§7.18 Repeated Integrals of the Complementary Error Function

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

§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

Keywords:: repeated integrals of the complementary error function
Permalink:: http://dlmf.nist.gov/7.18.i

7.18.1

$\mathop{\mathrm{i}^{{-1}}\mathrm{erfc}\/}\nolimits\!\left(z\right)=\frac{2}{\sqrt{\pi}}e^{{-z^{2}}},$

$\mathop{\mathrm{i}^{{0}}\mathrm{erfc}\/}\nolimits\!\left(z\right)=\mathop{\mathrm{erfc}\/}\nolimits z,$

Symbols:: $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, $e$ : base of exponential function, $\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(z\right)$ : 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

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

7.18.2 $\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(z\right)=\int _{z}^{\infty}\mathop{\mathrm{i}^{{n-1}}\mathrm{erfc}\/}\nolimits\!\left(t\right)dt=\frac{2}{\sqrt{\pi}}\int _{z}^{\infty}\frac{(t-z)^{n}}{n!}e^{{-t^{2}}}dt.$

Defines:: $\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(z\right)$ : repeated integrals of the complementary error function
Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $!$ : $n!$ : factorial, $\int$ : 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

§7.18(ii) Graphics

Notes:: These graphs were produced at NIST.
Keywords:: repeated integrals of the complementary error function
Permalink:: http://dlmf.nist.gov/7.18.ii

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(z\right)$ : repeated integrals of the complementary error function, $x$ : real variable and $n$ : nonnegative integer
Keywords:: repeated integrals of the complementary error function
Permalink:: http://dlmf.nist.gov/7.18.F1
Encodings:: pdf, png

§7.18(iii) Properties

Notes:: See Hartree (1936).
Keywords:: derivatives, repeated integrals of the complementary error function
Permalink:: http://dlmf.nist.gov/7.18.iii

7.18.3 $\frac{d}{dz}\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(z\right)=-\mathop{\mathrm{i}^{{n-1}}\mathrm{erfc}\/}\nolimits\!\left(z\right),$ $n=0,1,2,\dots$ ,

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(z\right)$ : 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

7.18.4 $\frac{{d}^{n}}{{dz}^{n}}\left(e^{{z^{2}}}\mathop{\mathrm{erfc}\/}\nolimits z\right)=(-1)^{n}2^{n}n!e^{{z^{2}}}\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(z\right),$ $n=0,1,2,\dots$ .

Symbols:: $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $e$ : base of exponential function, $!$ : $n!$ : factorial, $\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(z\right)$ : 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

7.18.5 $\frac{{d}^{2}W}{{dz}^{2}}+2z\frac{dW}{dz}-2nW=0,$ $W(z)=A\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(z\right)+B\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(-z\right)$ ,

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(z\right)$ : 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

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

7.18.6 $\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(z\right)=\sum _{{k=0}}^{\infty}\frac{(-1)^{k}z^{k}}{2^{{n-k}}k!\mathop{\Gamma\/}\nolimits\!\left(1+\frac{1}{2}(n-k)\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $!$ : $n!$ : factorial, $\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(z\right)$ : 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

7.18.7 $\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(z\right)=-\frac{z}{n}\mathop{\mathrm{i}^{{n-1}}\mathrm{erfc}\/}\nolimits\!\left(z\right)+\frac{1}{2n}\mathop{\mathrm{i}^{{n-2}}\mathrm{erfc}\/}\nolimits\!\left(z\right),$ $n=1,2,3,\dots$ .

Symbols:: $\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(z\right)$ : 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

§7.18(iv) Relations to Other Functions

Notes:: See Hartree (1936).
Permalink:: http://dlmf.nist.gov/7.18.iv

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

¶ Hermite Polynomials

Keywords:: Hermite polynomials, repeated integrals of the complementary error function

7.18.8 $(-1)^{n}\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(z\right)+\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(-z\right)=\frac{i^{{-n}}}{2^{{n-1}}n!}\mathop{H_{{n}}\/}\nolimits\!\left(iz\right).$

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $!$ : $n!$ : factorial, $\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(z\right)$ : 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

¶ Confluent Hypergeometric Functions

Keywords:: confluent hypergeometric functions, repeated integrals of the complementary error function

7.18.9 $\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(z\right)=e^{{-z^{2}}}\left(\frac{1}{2^{n}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}n+1\right)}\mathop{M\/}\nolimits\!\left(\tfrac{1}{2}n+\tfrac{1}{2},\tfrac{1}{2},z^{2}\right)-\frac{z}{2^{{n-1}}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}n+\tfrac{1}{2}\right)}\mathop{M\/}\nolimits\!\left(\tfrac{1}{2}n+1,\tfrac{3}{2},z^{2}\right)\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $e$ : base of exponential function, $\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(z\right)$ : 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

7.18.10 $\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(z\right)=\frac{e^{{-z^{2}}}}{2^{n}\sqrt{\pi}}\mathop{U\/}\nolimits\!\left(\tfrac{1}{2}n+\tfrac{1}{2},\tfrac{1}{2},z^{2}\right).$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $e$ : base of exponential function, $\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(z\right)$ : repeated integrals of the complementary error function, $z$ : complex variable and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/7.18.E10
Encodings:: TeX, pMML, png

¶ Parabolic Cylinder Functions

Keywords:: parabolic cylinder functions, repeated integrals of the complementary error function

7.18.11 $\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(z\right)=\frac{e^{{-z^{2}/2}}}{\sqrt{2^{{n-1}}\pi}}\mathop{U\/}\nolimits\!\left(n+\tfrac{1}{2},z\sqrt{2}\right).$

Symbols:: $e$ : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(z\right)$ : 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

¶ Probability Functions

Keywords:: parabolic cylinder functions, probability functions, repeated integrals of the complementary error function

7.18.12 $\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(z\right)=\frac{1}{\sqrt{2^{{n-1}}\pi}}\mathop{\mathit{Hh}_{{n}}\/}\nolimits\!\left(\sqrt{2}z\right).$

Defines:: $\mathop{\mathit{Hh}_{{n}}\/}\nolimits\!\left(z\right)$ : probability function
Symbols:: $\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(z\right)$ : 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 Jeffreys and Jeffreys (1956, §§23.081–23.09).

§7.18(v) Continued Fraction

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

7.18.13 $\frac{\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(z\right)}{\mathop{\mathrm{i}^{{n-1}}\mathrm{erfc}\/}\nolimits\!\left(z\right)}=\cfrac{1/2}{z+\cfrac{(n+1)/2}{z+\cfrac{(n+2)/2}{z+}}}\cdots,$ $\realpart{z}>0$ .

Symbols:: $\realpart{}$ : real part, $\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(z\right)$ : 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

§7.18(vi) Asymptotic Expansion

Notes:: See Hartree (1936).
Keywords:: repeated integrals of the complementary error function
Permalink:: http://dlmf.nist.gov/7.18.vi

7.18.14 $\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(z\right)\sim\frac{2}{\sqrt{\pi}}\frac{e^{{-z^{2}}}}{(2z)^{{n+1}}}\sum _{{m=0}}^{\infty}\frac{(-1)^{m}(2m+n)!}{n!m!(2z)^{{2m}}},$ $z\to\infty$ , $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{3}{4}\pi-\delta(<\tfrac{3}{4}\pi)$ .

Symbols:: $e$ : base of exponential function, $!$ : $n!$ : factorial, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\mathop{\mathrm{i}^{{n}}\mathrm{erfc}\/}\nolimits\!\left(z\right)$ : repeated integrals of the complementary error function, $\sim$ : asymptotic equality, $z$ : complex variable and $n$ : nonnegative integer
A&S Ref:: 7.2.14
Permalink:: http://dlmf.nist.gov/7.18.E14
Encodings:: TeX, pMML, png