error functions

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21: 7.7 Integral Representations

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§7.7(i) Error Functions and Dawson’s Integral

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7.7.1 $$\mathrm{erfc}z=\frac{2}{\pi }{\mathrm{e}}^{-z^2}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-z^2{t}^2}}{t^2+1}dt,$$ $|\mathrm{ph}z|\le \frac{1}{4}\pi$,

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Symbols:

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

A&S Ref:

7.4.11 (in different form)

Referenced by:

§7.11, §7.7(i)

Permalink:

http://dlmf.nist.gov/7.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(i), §7.7 and Ch.7

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7.7.5 $${\int }_0^1\frac{{\mathrm{e}}^{-a{t}^2}}{t^2+1}dt=\frac{\pi }{4}{\mathrm{e}}^a\left(1-{(\mathrm{erf}\sqrt{a})}^2\right),$$ $\mathrm{\Re }a>0$.

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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 $\mathrm{\Re }$: real part

A&S Ref:

7.4.12

Referenced by:

§7.7(i)

Permalink:

http://dlmf.nist.gov/7.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(i), §7.7 and Ch.7

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7.7.7 $${\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{-a^2{t}^2-(b^2/t^2)}dt=\frac{\sqrt{\pi }}{4a}\left({\mathrm{e}}^{2ab}\mathrm{erfc}\left(ax+(b/x)\right)+{\mathrm{e}}^{-2ab}\mathrm{erfc}\left(ax-(b/x)\right)\right),$$ $x>0$, .

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Symbols:

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

A&S Ref:

7.4.33 (in different form)

Referenced by:

§7.7(i), §7.8

Permalink:

http://dlmf.nist.gov/7.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(i), §7.7 and Ch.7

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7.7.9 $${\int }_0^x\mathrm{erf}tdt=x\mathrm{erf}x+\frac{1}{\sqrt{\pi }}\left({\mathrm{e}}^{-x^2}-1\right).$$

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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 $x$: real variable

A&S Ref:

7.4.35 (in different form)

Referenced by:

§7.7(i)

Permalink:

http://dlmf.nist.gov/7.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(i), §7.7 and Ch.7

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22: 7.20 Mathematical Applications

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§7.20(i) Asymptotics

►For applications of the complementary error function in uniform asymptotic approximations of integrals—saddle point coalescing with a pole or saddle point coalescing with an endpoint—see Wong (1989, Chapter 7), Olver (1997b, Chapter 9), and van der Waerden (1951). ►The complementary error function also plays a ubiquitous role in constructing exponentially-improved asymptotic expansions and providing a smooth interpretation of the Stokes phenomenon; see §§2.11(iii) and 2.11(iv). … ►

§7.20(iii) Statistics

… ►For applications in statistics and probability theory, also for the role of the normal distribution functions (the error functions and probability integrals) in the asymptotics of arbitrary probability density functions, see Johnson et al. (1994, Chapter 13) and Patel and Read (1982, Chapters 2 and 3).

23: 7 Error Functions, Dawson’s and Fresnel Integrals

Chapter 7 Error Functions, Dawson’s and Fresnel Integrals

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24: 12.7 Relations to Other Functions

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§12.7(ii) Error Functions, Dawson’s Integral, and Probability Function

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12.7.5 $$U(\frac{1}{2},z)=D_{-1}\left(z\right)=\sqrt{\frac{1}{2}\pi }{\mathrm{e}}^{\frac{1}{4}{z}^2}\mathrm{erfc}\left(z/\sqrt{2}\right),$$

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Symbols:

$D_{\nu }\left(z\right)$: parabolic cylinder function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function and $z$: complex variable

Referenced by:

§12.11(ii)

Permalink:

http://dlmf.nist.gov/12.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §12.7(ii), §12.7 and Ch.12

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12.7.6 $$U(n+\frac{1}{2},z)=D_{-n-1}\left(z\right)=\sqrt{\frac{\pi }{2}}\frac{{(-1)}^n}{n!}{\mathrm{e}}^{-\frac{1}{4}{z}^2}\frac{d^n\left({\mathrm{e}}^{\frac{1}{2}{z}^2}\mathrm{erfc}\left(z/\sqrt{2}\right)\right)}{{dz}^n},$$ $n=0,1,2,\mathrm{\dots }$,

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Symbols:

$D_{\nu }\left(z\right)$: parabolic cylinder function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $U(a,z)$: parabolic cylinder function, $z$: complex variable and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/12.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §12.7(ii), §12.7 and Ch.12

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12.7.7 $$U(n+\frac{1}{2},z)={\mathrm{e}}^{\frac{1}{4}{z}^2}{\mathrm{𝐻ℎ}}_n\left(z\right)=\sqrt{\pi }{\mathrm{\hspace{0.17em}2}}^{\frac{1}{2}(n-1)}{\mathrm{e}}^{\frac{1}{4}{z}^2}{\mathrm{i}}^n\mathrm{erfc}\left(z/\sqrt{2}\right),$$ $n=-1,0,1,\mathrm{\dots }$.

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Symbols:

${\mathrm{𝐻ℎ}}_n\left(z\right)$: probability function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§12.7(ii)

Permalink:

http://dlmf.nist.gov/12.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §12.7(ii), §12.7 and Ch.12

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25: 8.4 Special Values

… ►For $\mathrm{erf}\left(z\right)$, $\mathrm{erfc}\left(z\right)$, and $F\left(z\right)$, see §§7.2(i), 7.2(ii). … ►

8.4.1 $$\gamma (\frac{1}{2},z^2)=2{\int }_0^z{\mathrm{e}}^{-t^2}dt=\sqrt{\pi }\mathrm{erf}\left(z\right),$$

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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, $\gamma (a,z)$: incomplete gamma function, $\int$: integral and $z$: complex variable

A&S Ref:

6.5.16

Permalink:

http://dlmf.nist.gov/8.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.4 and Ch.8

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8.4.6 $$\mathrm{\Gamma }(\frac{1}{2},z^2)=2{\int }_z^{\mathrm{\infty }}{\mathrm{e}}^{-t^2}dt=\sqrt{\pi }\mathrm{erfc}\left(z\right).$$

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Symbols:

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

A&S Ref:

6.5.17

Referenced by:

§8.4

Permalink:

http://dlmf.nist.gov/8.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §8.4 and Ch.8

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8.4.14 $$Q(n+\frac{1}{2},z^2)=\mathrm{erfc}\left(z\right)+\frac{{\mathrm{e}}^{-z^2}}{\sqrt{\pi }}\sum _{k=1}^n\frac{z^{2k-1}}{{\left(\frac{1}{2}\right)}_k},$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $Q(a,z)$: normalized incomplete gamma function, $z$: complex variable, $k$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§8.4

Permalink:

http://dlmf.nist.gov/8.4.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §8.4 and Ch.8

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26: 7.8 Inequalities

§7.8 Inequalities

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7.8.8 $x>0$.

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Symbols:

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

Source:

Pólya (1949, (1.5))

Referenced by:

§7.8, Erratum (V1.0.17) for Equation (7.8.8)

Permalink:

http://dlmf.nist.gov/7.8.E8

Encodings:

pMML, png, TeX

Addition (effective with 1.0.17):

This inequality was added. It is Pólya (1949, (1.5)). Note that it is a special case of (8.10.11).

Suggested by Roberto Iacono

See also:

Annotations for §7.8 and Ch.7

27: 12.13 Sums

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12.13.6 $$n!U(n+\frac{1}{2},z)={\mathrm{i}}^n{\mathrm{e}}^{-\frac{1}{2}{z}^2}\mathrm{erfc}(z/\sqrt{2})U(-n-\frac{1}{2},\mathrm{i}z)+\sum _{m=1}^{\lfloor \frac{1}{2}n+\frac{1}{2}\rfloor }U(2m-n-\frac{1}{2},z),$$ $n=0,1,2,\mathrm{\dots }.$

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Symbols:

$\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\lfloor x\rfloor$: floor of $x$ , $\mathrm{i}$: imaginary unit, $U(a,z)$: parabolic cylinder function, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§12.13(i)

Permalink:

http://dlmf.nist.gov/12.13.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §12.13(i), §12.13 and Ch.12

►For $\mathrm{erfc}$ see §7.2(i). …

28: 8.12 Uniform Asymptotic Expansions for Large Parameter

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8.12.3 $$P(a,z)=\frac{1}{2}\mathrm{erfc}\left(-\eta \sqrt{a/2}\right)-S(a,\eta ),$$

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Symbols:

$\mathrm{erfc}z$: complementary error function, $P(a,z)$: normalized incomplete gamma function, $z$: complex variable, $a$: parameter, $\eta (\lambda )$ and $S(a,\eta )$: function

Permalink:

http://dlmf.nist.gov/8.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.12 and Ch.8

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8.12.4 $$Q(a,z)=\frac{1}{2}\mathrm{erfc}\left(\eta \sqrt{a/2}\right)+S(a,\eta ),$$

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Symbols:

$\mathrm{erfc}z$: complementary error function, $Q(a,z)$: normalized incomplete gamma function, $z$: complex variable, $a$: parameter, $\eta (\lambda )$ and $S(a,\eta )$: function

Referenced by:

§8.12

Permalink:

http://dlmf.nist.gov/8.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §8.12 and Ch.8

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$d(\pm \chi )=\sqrt{\frac{1}{2}\pi }{\mathrm{e}}^{{\chi }^2/2}\mathrm{erfc}\left(\pm \chi /\sqrt{2}\right),$

… ►For other uniform asymptotic approximations of the incomplete gamma functions in terms of the function $\mathrm{erfc}$ see Paris (2002b) and Dunster (1996a). … ►These expansions involve the inverse error function $\mathrm{inverfc}\left(x\right)$ (§7.17), and are uniform with respect to $q\in [0,1]$. …

29: Software Index

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	Open Source													With Book						Commercial
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7 Error Functions, Dawson’s and Fresnel Integrals
7.25(ii) $\mathrm{erf}x$, $\mathrm{erfc}x$, ${\mathrm{i}}^n\mathrm{erfc}\left(x\right)$, $x\in ℝ$	✓		✓	✓	✓	✓	✓	✓	✓		✓	✓	✓	✓	✓	✓	✓		✓	✓	✓	✓	✓	✓	NMS
7.25(iii) $\mathrm{erf}z$, $\mathrm{erfc}z$, $w\left(z\right)$, $z\in ℂ$	✓								✓			a	✓	✓							✓	✓	✓
…

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30: 13.18 Relations to Other Functions

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§13.18(ii) Incomplete Gamma Functions

… ►Special cases are the error functions ►

13.18.6 $$M_{-\frac{1}{4},\frac{1}{4}}\left(z^2\right)=\frac{1}{2}{\mathrm{e}}^{\frac{1}{2}{z}^2}\sqrt{\pi z}\mathrm{erf}\left(z\right),$$

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Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.18.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.18(ii), §13.18 and Ch.13

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13.18.7 $$W_{-\frac{1}{4},\pm \frac{1}{4}}\left(z^2\right)={\mathrm{e}}^{\frac{1}{2}{z}^2}\sqrt{\pi z}\mathrm{erfc}\left(z\right).$$

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Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

Referenced by:

Erratum (V1.0.3) for Equation (13.18.7)

Permalink:

http://dlmf.nist.gov/13.18.E7

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.3):

Originally the left-hand side was given correctly as $W_{-\frac{1}{4},-\frac{1}{4}}\left(z^2\right)$; the equation is true also for $W_{-\frac{1}{4},+\frac{1}{4}}\left(z^2\right)$.

See also:

Annotations for §13.18(ii), §13.18 and Ch.13

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