# erf

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##### 1: 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 $\mathsf{V}\left(x,t\right)$. Alternative notations are $Q(z)=\tfrac{1}{2}\operatorname{erfc}\left(z/\sqrt{2}\right)$, $P(z)=\Phi(z)=\tfrac{1}{2}\operatorname{erfc}\left(-z/\sqrt{2}\right)$, $\operatorname{Erf}z=\tfrac{1}{2}\sqrt{\pi}\operatorname{erf}z$, $\operatorname{Erfi}z=e^{z^{2}}F\left(z\right)$, $C_{1}(z)=C\left(\sqrt{2/\pi}z\right)$, $S_{1}(z)=S\left(\sqrt{2/\pi}z\right)$, $C_{2}(z)=C\left(\sqrt{2z/\pi}\right)$, $S_{2}(z)=S\left(\sqrt{2z/\pi}\right)$. …
##### 2: 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.

• Luke (1969b, pp. 323–324) covers $\frac{1}{2}\sqrt{\pi}\operatorname{erf}x$ and $e^{x^{2}}F\left(x\right)$ for $-3\leq x\leq 3$ (the Chebyshev coefficients are given to 20D); $\sqrt{\pi}xe^{x^{2}}\operatorname{erfc}x$ and $2xF\left(x\right)$ for $x\geq 3$ (the Chebyshev coefficients are given to 20D and 15D, respectively). Coefficients for the Fresnel integrals are given on pp. 328–330 (20D).

• 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).

• 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.

##### 4: 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.

• Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of $\operatorname{erf}z$, $x\in[0,5]$, $y=0.5(.5)3$, 7D and 8D, respectively; the real and imaginary parts of $\int_{x}^{\infty}e^{\pm\mathrm{i}t^{2}}\,\mathrm{d}t$, $(1/\sqrt{\pi})e^{\mp\mathrm{i}(x^{2}+(\pi/4))}\int_{x}^{\infty}e^{\pm\mathrm{i% }t^{2}}\,\mathrm{d}t$, $x=0(.5)20(1)25$, 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

• 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.

• ##### 5: 7.6 Series Expansions
7.6.8 $\operatorname{erf}z=\frac{2z}{\sqrt{\pi}}\sum_{n=0}^{\infty}(-1)^{n}\left({% \mathsf{i}^{(1)}_{2n}}\left(z^{2}\right)-{\mathsf{i}^{(1)}_{2n+1}}\left(z^{2}% \right)\right),$
7.6.9 $\operatorname{erf}\left(az\right)=\frac{2z}{\sqrt{\pi}}e^{(\frac{1}{2}-a^{2})z% ^{2}}\sum_{n=0}^{\infty}T_{2n+1}\left(a\right){\mathsf{i}^{(1)}_{n}}\left(% \tfrac{1}{2}z^{2}\right),$ $-1\leq a\leq 1$.
##### 6: 7.2 Definitions
7.2.1 $\operatorname{erf}z=\frac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^{2}}\,\mathrm{d}t,$
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. …
$\lim_{z\to\infty}\operatorname{erf}z=1,$
##### 7: 7.10 Derivatives
7.10.1 $\frac{{\mathrm{d}}^{n+1}\operatorname{erf}z}{{\mathrm{d}z}^{n+1}}=(-1)^{n}% \frac{2}{\sqrt{\pi}}H_{n}\left(z\right)e^{-z^{2}},$ $n=0,1,2,\dots$.
##### 8: 7.3 Graphics Figure 7.3.5: | erf ⁡ ( x + i ⁢ y ) | , − 3 ≤ x ≤ 3 , − 3 ≤ y ≤ 3 . … Magnify 3D Help
##### 9: 7.4 Symmetry
7.4.1 $\operatorname{erf}\left(-z\right)=-\operatorname{erf}\left(z\right),$
##### 10: 7.13 Zeros
###### §7.13(i) Zeros of $\operatorname{erf}z$
$\operatorname{erf}z$ has a simple zero at $z=0$, and in the first quadrant of $\mathbb{C}$ there is an infinite set of zeros $z_{n}=x_{n}+iy_{n}$, $n=1,2,3,\dots$, arranged in order of increasing absolute value. The other zeros of $\operatorname{erf}z$ are $-z_{n}$, $\overline{z}_{n}$, $-\overline{z}_{n}$. …