error functions

SearchAdvancedHelp

(0.009 seconds)

1—10 of 171 matching pages

1: 7.18 Repeated Integrals of the Complementary Error Function

§7.18 Repeated Integrals of the Complementary Error Function

… ►

${\mathrm{i}}^{-1}\mathrm{erfc}\left(z\right)={\displaystyle \frac{2}{\sqrt{\pi }}}{\mathrm{e}}^{-z^2},$

►

${\mathrm{i}}^0\mathrm{erfc}\left(z\right)=\mathrm{erfc}z,$

… ►

7.18.2 $${\mathrm{i}}^n\mathrm{erfc}\left(z\right)={\int }_z^{\mathrm{\infty }}{\mathrm{i}}^{n-1}\mathrm{erfc}\left(t\right)dt=\frac{2}{\sqrt{\pi }}{\int }_z^{\mathrm{\infty }}\frac{{(t-z)}^n}{n!}{\mathrm{e}}^{-t^2}dt.$$

ⓘ

Defines:

${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: 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:

pMML, png, TeX

See also:

Annotations for §7.18(i), §7.18 and Ch.7

… ►

7.18.7 $${\mathrm{i}}^n\mathrm{erfc}\left(z\right)=-\frac{z}{n}{\mathrm{i}}^{n-1}\mathrm{erfc}\left(z\right)+\frac{1}{2n}{\mathrm{i}}^{n-2}\mathrm{erfc}\left(z\right),$$ $n=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: 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:

pMML, png, TeX

See also:

Annotations for §7.18(iii), §7.18 and Ch.7

…

2: 7.2 Definitions

… ►

§7.2(i) Error Functions

… ►

7.2.2 $$\mathrm{erfc}z=\frac{2}{\sqrt{\pi }}{\int }_z^{\mathrm{\infty }}{\mathrm{e}}^{-t^2}dt=1-\mathrm{erf}z,$$

ⓘ

Defines:

$\mathrm{erfc}z$: complementary error function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $z$: complex variable

A&S Ref:

7.1.2

Referenced by:

§7.5, §7.7(i)

Permalink:

http://dlmf.nist.gov/7.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §7.2(i), §7.2 and Ch.7

… ► $\mathrm{erf}z$, $\mathrm{erfc}z$, and $w\left(z\right)$ are entire functions of $z$, as is $F\left(z\right)$ in the next subsection. ►

Values at Infinity

►

$\underset{z\to \mathrm{\infty }}{lim}\mathrm{erf}z=1,$

…

3: 7.25 Software

… ►

§7.25(ii) $\mathrm{erf}x$, $\mathrm{erfc}x$, ${\mathrm{i}}^n\mathrm{erfc}\left(x\right)$, $x\in ℝ$

… ►

§7.25(iii) $\mathrm{erf}z$, $\mathrm{erfc}z$, $w\left(z\right)$, $z\in ℂ$

…

4: 7.10 Derivatives

§7.10 Derivatives

►

7.10.1 $$\frac{d^{n+1}\mathrm{erf}z}{{dz}^{n+1}}={(-1)}^n\frac{2}{\sqrt{\pi }}{H}_n\left(z\right){\mathrm{e}}^{-z^2},$$ $n=0,1,2,\mathrm{\dots }$.

ⓘ

Symbols:

$H_n\left(x\right)$: Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.1.19

Permalink:

http://dlmf.nist.gov/7.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.10 and Ch.7

… ►

7.10.2 $$w^{\prime }\left(z\right)=-2zw\left(z\right)+(2\mathrm{i}/\sqrt{\pi }),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

7.1.20

Permalink:

http://dlmf.nist.gov/7.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §7.10 and Ch.7

►

7.10.3 $$w^{(n+2)}\left(z\right)+2z{w}^{(n+1)}\left(z\right)+2(n+1)w^{(n)}\left(z\right)=0,$$ $n=0,1,2,\mathrm{\dots }$.

ⓘ

Symbols:

$w\left(z\right)$: Faddeeva (or Faddeyeva) function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.1.20

Permalink:

http://dlmf.nist.gov/7.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §7.10 and Ch.7

…

5: 7.24 Approximations

… ►

Hastings (1955) gives several minimax polynomial and rational approximations for $\mathrm{erf}x$, $\mathrm{erfc}x$ and the auxiliary functions $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$.

►

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

… ►

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

►

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

… ►

Luke (1969b, vol. 2, pp. 422–435) gives main diagonal Padé approximations for $F\left(z\right)$, $\mathrm{erf}z$, $\mathrm{erfc}z$, $C\left(z\right)$, 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 $\mathrm{erf}z$; the complementary error functions $\mathrm{erfc}z$ and $w\left(z\right)$; Dawson’s integral $F\left(z\right)$; the Fresnel integrals $ℱ\left(z\right)$, $C\left(z\right)$, and $S\left(z\right)$; the Goodwin–Staton integral $G\left(z\right)$; the repeated integrals of the complementary error function ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$; the Voigt functions $𝖴(x,t)$ and $𝖵(x,t)$. ►Alternative notations are $Q(z)=\frac{1}{2}\mathrm{erfc}\left(z/\sqrt{2}\right)$, $P(z)=\mathrm{\Phi }(z)=\frac{1}{2}\mathrm{erfc}\left(-z/\sqrt{2}\right)$, $\mathrm{Erf}z=\frac{1}{2}\sqrt{\pi }\mathrm{erf}z$, $\mathrm{Erfi}z={\mathrm{e}}^{z^2}F\left(z\right)$, $C_1(z)=C\left(\sqrt{2/\pi }z\right)$, $S_1(z)=S\left(\sqrt{2/\pi }z\right)$, $C_2(z)=C\left(\sqrt{2z/\pi }\right)$, $S_2(z)=S\left(\sqrt{2z/\pi }\right)$. … ►

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 ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$. 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 $\mathrm{erfc}x$ and $\mathrm{erfc}\left(10x\right)$, $-3\le x\le 3$. Magnify

… ►

►

►

Figure 7.3.5: $|\mathrm{erf}\left(x+\mathrm{i}y\right)|$, $-3\le x\le 3$, $-3\le y\le 3$. … Magnify 3D Help

►

►

►

Figure 7.3.6: $|\mathrm{erfc}\left(x+\mathrm{i}y\right)|$, $-3\le x\le 3$, $-3\le y\le 3$. … Magnify 3D Help

9: 7.17 Inverse Error Functions

§7.17 Inverse Error Functions

… ►The inverses of the functions $x=\mathrm{erf}y$, $x=\mathrm{erfc}y$, $y\in ℝ$, are denoted by ►

$y=\mathrm{inverf}x,$

►

$y=\mathrm{inverfc}x,$

… ►

§7.17(iii) Asymptotic Expansion of $\mathrm{inverfc}x$ for Small $x$

…

10: 7.23 Tables

… ►

Abramowitz and Stegun (1964, Chapter 7) includes $\mathrm{erf}x$, $(2/\sqrt{\pi }){\mathrm{e}}^{-x^2}$, $x\in [0,2]$, 10D; $(2/\sqrt{\pi }){\mathrm{e}}^{-x^2}$, $x\in [2,10]$, 8S; $x{\mathrm{e}}^{x^2}\mathrm{erfc}x$, $x^{-2}\in [0,0.25]$, 7D; $2^n\mathrm{\Gamma }\left(\frac{1}{2}n+1\right){\mathrm{i}}^n\mathrm{erfc}\left(x\right)$, $n=1(1)6,10,11$, $x\in [0,5]$, 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 $(2/\sqrt{\pi }){\mathrm{e}}^{-x^2}$, $\mathrm{erf}x$, $x=0(.02)1(.04)3$, 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 $\mathrm{erf}z$ and $w\left(z\right)$ (the table on page 406 of this reference is for $w\left(z\right)$, not for $\mathrm{erfc}z$), 11S.

►

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