# error functions

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##### 1: 7.18 Repeated Integrals of the Complementary Error Function
###### §7.18 Repeated Integrals of the Complementary ErrorFunction
$\mathop{\mathrm{i}^{-1}\mathrm{erfc}}\left(z\right)=\frac{2}{\sqrt{\pi}}e^{-z^% {2}},$
$\mathop{\mathrm{i}^{0}\mathrm{erfc}}\left(z\right)=\operatorname{erfc}z,$
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: 7.2 Definitions
###### §7.2(i) ErrorFunctions
7.2.2 $\operatorname{erfc}z=\frac{2}{\sqrt{\pi}}\int_{z}^{\infty}e^{-t^{2}}\,\mathrm{% d}t=1-\operatorname{erf}z,$
$\operatorname{erf}z$, $\operatorname{erfc}z$, and $w\left(z\right)$ are entire functions of $z$, as is $F\left(z\right)$ in the next subsection.
###### Values at Infinity
$\lim_{z\to\infty}\operatorname{erf}z=1,$
##### 4: 7.10 Derivatives
###### §7.10 Derivatives
7.10.1 $\frac{{\mathrm{d}}^{n+1}\operatorname{erf}z}{{\mathrm{d}z}^{n+1}}=(-1)^{n}% \frac{2}{\sqrt{\pi}}H_{n}\left(z\right)e^{-z^{2}},$ $n=0,1,2,\dots$.
7.10.3 ${{w}^{(n+2)}\left(z\right)+2z{w}^{(n+1)}\left(z\right)+2(n+1){w}^{(n)}\left(z% \right)=0},$ $n=0,1,2,\dots$.
##### 5: 7.24 Approximations
• Hastings (1955) gives several minimax polynomial and rational approximations for $\operatorname{erf}x$, $\operatorname{erfc}x$ and the auxiliary functions $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$.

• Cody (1969) provides minimax rational approximations for $\operatorname{erf}x$ and $\operatorname{erfc}x$. The maximum relative precision is about 20S.

• Schonfelder (1978) gives coefficients of Chebyshev expansions for $x^{-1}\operatorname{erf}x$ on $0\leq x\leq 2$, for $xe^{x^{2}}\operatorname{erfc}x$ on $[2,\infty)$, and for $e^{x^{2}}\operatorname{erfc}x$ on $[0,\infty)$ (30D).

• Shepherd and Laframboise (1981) gives coefficients of Chebyshev series for $(1+2x)e^{x^{2}}\operatorname{erfc}x$ on $(0,\infty)$ (22D).

• Luke (1969b, vol. 2, pp. 422–435) gives main diagonal Padé approximations for $F\left(z\right)$, $\operatorname{erf}z$, $\operatorname{erfc}z$, $C\left(z\right)$, and $S\left(z\right)$; approximate errors are given for a selection of $z$-values.

• ##### 6: 7.1 Special Notation
Unless otherwise noted, primes indicate derivatives with respect to the argument. 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 $\mathsf{V}\left(x,t\right)$. Alternative notations are $Q(z)=\tfrac{1}{2}\operatorname{erfc}\left(z/\sqrt{2}\right)$, $P(z)=\Phi(z)=\tfrac{1}{2}\operatorname{erfc}\left(-z/\sqrt{2}\right)$, $\operatorname{Erf}z=\tfrac{1}{2}\sqrt{\pi}\operatorname{erf}z$, $\operatorname{Erfi}z=e^{z^{2}}F\left(z\right)$, $C_{1}(z)=C\left(\sqrt{2/\pi}z\right)$, $S_{1}(z)=S\left(\sqrt{2/\pi}z\right)$, $C_{2}(z)=C\left(\sqrt{2z/\pi}\right)$, $S_{2}(z)=S\left(\sqrt{2z/\pi}\right)$. …
##### 7: 7.21 Physical Applications
###### §7.21 Physical Applications
The error functions, Fresnel integrals, and related functions occur in a variety of physical applications. … Carslaw and Jaeger (1959) gives many applications and points out the importance of the repeated integrals of the complementary error function $\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)$. Fried and Conte (1961) mentions the role of $w\left(z\right)$ in the theory of linearized waves or oscillations in a hot plasma; $w\left(z\right)$ is called the plasma dispersion function or Faddeeva (or Faddeyeva) function; see Faddeeva and Terent’ev (1954). …
##### 8: 7.3 Graphics Figure 7.3.1: Complementary error functions erfc ⁡ x and erfc ⁡ ( 10 ⁢ x ) , − 3 ≤ x ≤ 3 . Magnify Figure 7.3.5: | erf ⁡ ( x + i ⁢ y ) | , − 3 ≤ x ≤ 3 , − 3 ≤ y ≤ 3 . … Magnify 3D Help Figure 7.3.6: | erfc ⁡ ( x + i ⁢ y ) | , − 3 ≤ x ≤ 3 , − 3 ≤ y ≤ 3 . … Magnify 3D Help
##### 9: 7.17 Inverse Error Functions
###### §7.17 Inverse ErrorFunctions
The inverses of the functions $x=\operatorname{erf}y$, $x=\operatorname{erfc}y$, $y\in\mathbb{R}$, are denoted by
$y=\operatorname{inverf}x,$
$y=\operatorname{inverfc}x,$
##### 10: 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.

• Abramowitz and Stegun (1964, Chapter 7) includes $w\left(z\right)$, $x=0(.1)3.9$, $y=0(.1)3$, 6D.

• Fettis et al. (1973) gives the first 100 zeros of $\operatorname{erf}z$ and $w\left(z\right)$ (the table on page 406 of this reference is for $w\left(z\right)$, not for $\operatorname{erfc}z$), 11S.

• Zhang and Jin (1996, p. 642) includes the first 10 zeros of $\operatorname{erf}z$, 9D; the first 25 distinct zeros of $C\left(z\right)$ and $S\left(z\right)$, 8S.