# repeated integrals of the complementary error function

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##### 1: 7.18 Repeated Integrals of the Complementary Error Function
###### §7.18 RepeatedIntegrals of the ComplementaryErrorFunction
$\mathop{\mathrm{i}^{-1}\mathrm{erfc}}\left(z\right)=\frac{2}{\sqrt{\pi}}e^{-z^% {2}},$
$\mathop{\mathrm{i}^{0}\mathrm{erfc}}\left(z\right)=\operatorname{erfc}z,$
7.18.2 $\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)=\int_{z}^{\infty}\mathop{% \mathrm{i}^{n-1}\mathrm{erfc}}\left(t\right)\,\mathrm{d}t=\frac{2}{\sqrt{\pi}}% \int_{z}^{\infty}\frac{(t-z)^{n}}{n!}e^{-t^{2}}\,\mathrm{d}t.$
7.18.7 $\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)=-\frac{z}{n}\mathop{\mathrm% {i}^{n-1}\mathrm{erfc}}\left(z\right)+\frac{1}{2n}\mathop{\mathrm{i}^{n-2}% \mathrm{erfc}}\left(z\right),$ $n=1,2,3,\dots$.
##### 2: 7.22 Methods of Computation
###### §7.22(iii) RepeatedIntegrals of the ComplementaryErrorFunction
The recursion scheme given by (7.18.1) and (7.18.7) can be used for computing $\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(x\right)$. …
##### 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 $\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)$. …
##### 5: 7.1 Special Notation
The main functions treated in this chapter are the error function $\operatorname{erf}z$; the complementary error functions $\operatorname{erfc}z$ and $w\left(z\right)$; Dawson’s integral $F\left(z\right)$; the Fresnel integrals $\mathcal{F}\left(z\right)$, $C\left(z\right)$, and $S\left(z\right)$; the Goodwin–Staton integral $G\left(z\right)$; the repeated integrals of the complementary error function $\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)$; the Voigt functions $\mathsf{U}\left(x,t\right)$ and $\mathsf{V}\left(x,t\right)$. …
##### 6: 7.23 Tables
• Abramowitz and Stegun (1964, Chapter 7) includes $\operatorname{erf}x$, $(2/\sqrt{\pi})e^{-x^{2}}$, $x\in[0,2]$, 10D; $(2/\sqrt{\pi})e^{-x^{2}}$, $x\in[2,10]$, 8S; $xe^{x^{2}}\operatorname{erfc}x$, $x^{-2}\in[0,0.25]$, 7D; $2^{n}\Gamma\left(\frac{1}{2}n+1\right)\mathop{\mathrm{i}^{n}\mathrm{erfc}}% \left(x\right)$, $n=1(1)6,10,11$, $x\in[0,5]$, 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.

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

• ##### 7: 12.7 Relations to Other Functions
###### §12.7(ii) ErrorFunctions, Dawson’s Integral, and Probability Function
12.7.7 $U\left(n+\tfrac{1}{2},z\right)=e^{\frac{1}{4}z^{2}}\mathit{Hh}_{n}\left(z% \right)=\sqrt{\pi}\,2^{\frac{1}{2}(n-1)}e^{\frac{1}{4}z^{2}}\mathop{\mathrm{i}% ^{n}\mathrm{erfc}}\left(z/\sqrt{2}\right),$ $n=-1,0,1,\dots$.
##### 8: Software Index
 NMS ✓ Open Source With Book Commercial … 7.25(ii) $\operatorname{erf}x$, $\operatorname{erfc}x$, $\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(x\right)$, $x\in\mathbb{R}$ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ …
##### 9: 2.4 Contour Integrals
If, in addition, the corresponding integrals with $Q$ and $F$ replaced by their derivatives $Q^{(j)}$ and $F^{(j)}$, $j=1,2,\dots,m$, 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. …