repeated integrals of the complementary error function

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1: 7.18 Repeated Integrals of the Complementary Error Function

§7.18 Repeated Integrals of the Complementary Error Function

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

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

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

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Defines:

${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: 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:

pMML, png, TeX

See also:

Annotations for §7.18(i), §7.18 and Ch.7

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

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Symbols:

${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: 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:

pMML, png, TeX

See also:

Annotations for §7.18(iii), §7.18 and Ch.7

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2: 7.22 Methods of Computation

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§7.22(iii) Repeated Integrals of the Complementary Error Function

►The recursion scheme given by (7.18.1) and (7.18.7) can be used for computing ${\mathrm{i}}^n\mathrm{erfc}\left(x\right)$. …

3: 7.25 Software

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§7.25(ii) $\mathrm{erf}x$, $\mathrm{erfc}x$, ${\mathrm{i}}^n\mathrm{erfc}\left(x\right)$, $x\in ℝ$

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4: 7.21 Physical Applications

§7.21 Physical Applications

… ►Carslaw and Jaeger (1959) gives many applications and points out the importance of the repeated integrals of the complementary error function ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$. …

5: 7.1 Special Notation

… ►The main functions treated in this chapter are the error function $\mathrm{erf}z$; the complementary error functions $\mathrm{erfc}z$ and $w\left(z\right)$; Dawson’s integral $F\left(z\right)$; the Fresnel integrals $ℱ\left(z\right)$, $C\left(z\right)$, and $S\left(z\right)$; the Goodwin–Staton integral $G\left(z\right)$; the repeated integrals of the complementary error function ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$; the Voigt functions $𝖴(x,t)$ and $𝖵(x,t)$. …

6: 7.23 Tables

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Abramowitz and Stegun (1964, Chapter 7) includes $\mathrm{erf}x$, $(2/\sqrt{\pi }){\mathrm{e}}^{-x^2}$, $x\in [0,2]$, 10D; $(2/\sqrt{\pi }){\mathrm{e}}^{-x^2}$, $x\in [2,10]$, 8S; $x{\mathrm{e}}^{x^2}\mathrm{erfc}x$, $x^{-2}\in [0,0.25]$, 7D; $2^n\mathrm{\Gamma }\left(\frac{1}{2}n+1\right){\mathrm{i}}^n\mathrm{erfc}\left(x\right)$, $n=1(1)6,10,11$, $x\in [0,5]$, 6S; $F\left(x\right)$, $x\in [0,2]$, 10D; $xF\left(x\right)$, $x^{-2}\in [0,0.25]$, 9D; $C\left(x\right)$, $S\left(x\right)$, $x\in [0,5]$, 7D; $\mathrm{f}\left(x\right)$, $\mathrm{g}\left(x\right)$, $x\in [0,1]$, $x^{-1}\in [0,1]$, 15D.

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Zhang and Jin (1996, pp. 637, 639) includes $(2/\sqrt{\pi }){\mathrm{e}}^{-x^2}$, $\mathrm{erf}x$, $x=0(.02)1(.04)3$, 8D; $C\left(x\right)$, $S\left(x\right)$, $x=0(.2)10(2)100(100)500$, 8D.

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7: 12.7 Relations to Other Functions

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§12.7(ii) Error Functions, Dawson’s Integral, and Probability Function

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12.7.7 $$U(n+\frac{1}{2},z)={\mathrm{e}}^{\frac{1}{4}{z}^2}{\mathrm{𝐻ℎ}}_n\left(z\right)=\sqrt{\pi }{\mathrm{\hspace{0.17em}2}}^{\frac{1}{2}(n-1)}{\mathrm{e}}^{\frac{1}{4}{z}^2}{\mathrm{i}}^n\mathrm{erfc}\left(z/\sqrt{2}\right),$$ $n=-1,0,1,\mathrm{\dots }$.

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Symbols:

${\mathrm{𝐻ℎ}}_n\left(z\right)$: probability function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§12.7(ii)

Permalink:

http://dlmf.nist.gov/12.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §12.7(ii), §12.7 and Ch.12

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8: Software Index

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	Open Source													With Book						Commercial
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7.25(ii) $\mathrm{erf}x$, $\mathrm{erfc}x$, ${\mathrm{i}}^n\mathrm{erfc}\left(x\right)$, $x\in ℝ$	✓		✓	✓	✓	✓	✓	✓	✓		✓	✓	✓	✓	✓	✓	✓		✓	✓	✓	✓	✓	✓	NMS
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9: 2.4 Contour Integrals

… ►If, in addition, the corresponding integrals with $Q$ and $F$ replaced by their derivatives $Q^{(j)}$ and $F^{(j)}$, $j=1,2,\mathrm{\dots },m$, converge uniformly, then by repeated integrations by parts … ►For error bounds see Boyd (1993). … ►Thus the right-hand side of (2.4.14) reduces to the error terms. … ►For a coalescing saddle point and a pole see Wong (1989, Chapter 7) and van der Waerden (1951); in this case the uniform approximants are complementary error functions. For a coalescing saddle point and endpoint see Olver (1997b, Chapter 9) and Wong (1989, Chapter 7); if the endpoint is an algebraic singularity then the uniform approximants are parabolic cylinder functions with fixed parameter, and if the endpoint is not a singularity then the uniform approximants are complementary error functions. …