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11: 7.5 Interrelations

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7.5.1 $$F\left(z\right)=\frac{1}{2}\mathrm{i}\sqrt{\pi }\left({\mathrm{e}}^{-z^2}-w\left(z\right)\right)=-\frac{1}{2}\mathrm{i}\sqrt{\pi }{\mathrm{e}}^{-z^2}\mathrm{erf}\left(\mathrm{i}z\right).$$

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Symbols:

$F\left(z\right)$: Dawson’s integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§7.5

Permalink:

http://dlmf.nist.gov/7.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.8 $$C\left(z\right)\pm \mathrm{i}S\left(z\right)=\frac{1}{2}(1\pm \mathrm{i})\mathrm{erf}\zeta .$$

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Symbols:

$C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\mathrm{erf}z$: error function, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta$: change of variable

Referenced by:

§7.5, §7.5, §7.6(i)

Permalink:

http://dlmf.nist.gov/7.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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12: 7.11 Relations to Other Functions

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7.11.1 $$\mathrm{erf}z=\frac{1}{\sqrt{\pi }}\gamma (\frac{1}{2},z^2),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $\gamma (a,z)$: incomplete gamma function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/7.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.11, §7.11 and Ch.7

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7.11.4 $$\mathrm{erf}z=\frac{2z}{\sqrt{\pi }}M(\frac{1}{2},\frac{3}{2},-z^2)=\frac{2z}{\sqrt{\pi }}{\mathrm{e}}^{-z^2}M(1,\frac{3}{2},z^2),$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer 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

A&S Ref:

7.1.21

Permalink:

http://dlmf.nist.gov/7.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §7.11, §7.11 and Ch.7

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13: 7.14 Integrals

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7.14.2 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\mathrm{erf}\left(bt\right)dt=\frac{1}{a}{\mathrm{e}}^{a^2/(4b^2)}\mathrm{erfc}\left(\frac{a}{2b}\right),$$ $\mathrm{\Re }a>0$, ,

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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 $x$ , $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.17

Referenced by:

§7.14(i)

Permalink:

http://dlmf.nist.gov/7.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(i), §7.14(i), §7.14 and Ch.7

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7.14.3 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\mathrm{erf}\sqrt{bt}dt=\frac{1}{a}\sqrt{\frac{b}{a+b}},$$ $\mathrm{\Re }a>0$, $\mathrm{\Re }b>0$,

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Symbols:

$dx$: differential of $x$ , $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.19

Referenced by:

§7.14(i)

Permalink:

http://dlmf.nist.gov/7.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(i), §7.14(i), §7.14 and Ch.7

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14: 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

15: 7.7 Integral Representations

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

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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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16: 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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17: 7.17 Inverse Error Functions

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

18: 13.18 Relations to Other Functions

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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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19: Software Index

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	Open Source													With Book						Commercial
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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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20: 13.6 Relations to Other Functions

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13.6.7 $$M(\frac{1}{2},\frac{3}{2},-z^2)=\frac{\sqrt{\pi }}{2z}\mathrm{erf}\left(z\right),$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function and $z$: complex variable

A&S Ref:

13.6.19

Referenced by:

§13.6(ii)

Permalink:

http://dlmf.nist.gov/13.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(ii), §13.6 and Ch.13

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