# repeated

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##### 2: 7.22 Methods of Computation
###### §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 $\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(x\right)$. …
##### 3: 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: 7.23 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.

• ##### 6: 12.7 Relations to Other Functions
###### §12.7(ii) Error Functions, Dawson’s Integral, and Probability Function
12.7.7 $U\left(n+\tfrac{1}{2},z\right)=e^{\frac{1}{4}z^{2}}\mathit{Hh}_{n}\left(z% \right)=\sqrt{\pi}\,2^{\frac{1}{2}(n-1)}e^{\frac{1}{4}z^{2}}\mathop{\mathrm{i}% ^{n}\mathrm{erfc}}\left(z/\sqrt{2}\right),$ $n=-1,0,1,\dots$.
##### 7: 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 $\mathsf{V}\left(x,t\right)$. …
##### 8: 27.15 Chinese Remainder Theorem
Even though the lengthy calculation is repeated four times, once for each modulus, most of it only uses five-digit integers and is accomplished quickly without overwhelming the machine’s memory. …
##### 9: 4.12 Generalized Logarithms and Exponentials
4.12.9 $\psi(x)=\ell+\underbrace{\ln\cdots\ln}_{\ell\text{ times}}x,$ $x>1$,
4.12.10 $0\leq\underbrace{\ln\cdots\ln}_{\ell\text{times}}x<1.$
##### 10: Bibliography G
• W. Gautschi (1977a) Evaluation of the repeated integrals of the coerror function. ACM Trans. Math. Software 3, pp. 240–252.
• W. Gautschi (1977b) Algorithm 521: Repeated integrals of the coerror function. ACM Trans. Math. Software 3, pp. 301–302.
• W. Gautschi (1961) Recursive computation of the repeated integrals of the error function. Math. Comp. 15 (75), pp. 227–232.
• W. Gautschi (2016) Algorithm 957: evaluation of the repeated integral of the coerror function by half-range Gauss-Hermite quadrature. ACM Trans. Math. Softw. 42 (1), pp. 9:1–9:10.