# §7.18 Repeated Integrals of the Complementary Error Function

## §7.18(i) Definition

 7.18.1 $\displaystyle\mathop{\mathrm{i}^{-1}\mathrm{erfc}}\left(z\right)$ $\displaystyle=\frac{2}{\sqrt{\pi}}e^{-z^{2}},$ $\displaystyle\mathop{\mathrm{i}^{0}\mathrm{erfc}}\left(z\right)$ $\displaystyle=\operatorname{erfc}z,$ ⓘ 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(iii) Properties

 7.18.3 $\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z% \right)=-\mathop{\mathrm{i}^{n-1}\mathrm{erfc}}\left(z\right),$ $n=0,1,2,\dots$,
 7.18.4 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{z^{2}}\operatorname{erfc}z% \right)=(-1)^{n}2^{n}n!e^{z^{2}}\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z% \right),$ $n=0,1,2,\dots$.
 7.18.5 $\frac{{\mathrm{d}}^{2}W}{{\mathrm{d}z}^{2}}+2z\frac{\mathrm{d}W}{\mathrm{d}z}-% 2nW=0,$ $W(z)=A\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)+B\mathop{\mathrm{i}^{% n}\mathrm{erfc}}\left(-z\right)$,

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

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

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

### Hermite Polynomials

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

### Confluent Hypergeometric Functions

 7.18.9 $\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)=e^{-z^{2}}\left(\frac{1}{2^% {n}\Gamma\left(\tfrac{1}{2}n+1\right)}M\left(\tfrac{1}{2}n+\tfrac{1}{2},\tfrac% {1}{2},z^{2}\right)-\frac{z}{2^{n-1}\Gamma\left(\tfrac{1}{2}n+\tfrac{1}{2}% \right)}M\left(\tfrac{1}{2}n+1,\tfrac{3}{2},z^{2}\right)\right),$
 7.18.10 $\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)=\frac{e^{-z^{2}}}{2^{n}% \sqrt{\pi}}U\left(\tfrac{1}{2}n+\tfrac{1}{2},\tfrac{1}{2},z^{2}\right).$

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

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

### Probability Functions

 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

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

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