DLMF: Untitled Document

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8.4.6

\Gamma\left(\tfrac{1}{2},z^{2}\right)=2\int_{z}^{\infty}e^{-t^{2}}\,\mathrm{d}% t=\sqrt{\pi}\operatorname{erfc}\left(z\right).

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : 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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7.2.2

\operatorname{erfc}z=\frac{2}{\sqrt{\pi}}\int_{z}^{\infty}e^{-t^{2}}\,\mathrm{% d}t=1-\operatorname{erf}z,

ⓘ

Defines:: $\operatorname{erfc}\NVar{z}$ : complementary error function
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{erf}\NVar{z}$ : 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

►

7.2.3

w\left(z\right)=e^{-z^{2}}\left(1+\frac{2i}{\sqrt{\pi}}\int_{0}^{z}e^{t^{2}}\,% \mathrm{d}t\right)=e^{-z^{2}}\operatorname{erfc}\left(-iz\right).

ⓘ

Defines:: $w\left(\NVar{z}\right)$ : Faddeeva (or Faddeyeva) function
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral and $z$ : complex variable
Keywords:: definition, Faddeeva (or Faddeyeva) function
A&S Ref:: 7.1.3
Referenced by:: §7.19(i), §7.5, Erratum (V1.0.17) for Equation (7.2.3)
Permalink:: http://dlmf.nist.gov/7.2.E3
Encodings:: pMML, png, TeX
Clarification (effective with 1.0.17):: Originally named as a complementary error function, this has been renamed as the Faddeeva (or Faddeyeva) function.
See also:: Annotations for §7.2(i), §7.2 and Ch.7

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12.7.7

U\left(n+\tfrac{1}{2},z\right)=e^{\frac{1}{4}z^{2}}\mathit{Hh}_{n}\left(z% \right)=\sqrt{\pi}\,2^{\frac{1}{2}(n-1)}e^{\frac{1}{4}z^{2}}\mathop{\mathrm{i}% ^{n}\mathrm{erfc}}\left(z/\sqrt{2}\right),

n=-1,0,1,\dots

.

ⓘ

Symbols:: $\mathit{Hh}_{\NVar{n}}\left(\NVar{z}\right)$ : probability function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $\mathop{\mathrm{i}^{\NVar{n}}\mathrm{erfc}}\left(\NVar{z}\right)$ : 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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3.5.42

\operatorname{erfc}\lambda=\frac{1}{2\pi\mathrm{i}}\int_{c-\mathrm{i}\infty}^{% c+\mathrm{i}\infty}e^{\zeta-2\lambda\sqrt{\zeta}}\frac{\,\mathrm{d}\zeta}{% \zeta},

c>0

,

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\lambda$ : parameter
A&S Ref:: 29.3.83 (in different form)
Referenced by:: §3.5(ix)
Permalink:: http://dlmf.nist.gov/3.5.E42
Encodings:: pMML, png, TeX
See also:: Annotations for §3.5(ix), §3.5(ix), §3.5 and Ch.3

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3.5.44

\operatorname{erfc}\lambda=\frac{1}{2\pi\mathrm{i}}\int_{c-\mathrm{i}\infty}^{% c+\mathrm{i}\infty}e^{\lambda^{2}(t-2\sqrt{t})}\frac{\,\mathrm{d}t}{t},

c>0

,

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\lambda$ : parameter
Permalink:: http://dlmf.nist.gov/3.5.E44
Encodings:: pMML, png, TeX
See also:: Annotations for §3.5(ix), §3.5(ix), §3.5 and Ch.3

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3.5.45

\operatorname{erfc}\lambda=\frac{e^{-\lambda^{2}}}{2\pi}\int_{-\pi}^{\pi}e^{-% \lambda^{2}{\tan}^{2}\left(\frac{1}{2}\theta\right)}\,\mathrm{d}\theta.

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\tan\NVar{z}$ : tangent function and $\lambda$ : parameter
Referenced by:: Table 3.5.20
Permalink:: http://dlmf.nist.gov/3.5.E45
Encodings:: pMML, png, TeX
See also:: Annotations for §3.5(ix), §3.5(ix), §3.5 and Ch.3

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… ►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)

. …

§7.8 Inequalities

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7.8.6

\int_{0}^{x}e^{at^{2}}\,\mathrm{d}t<\frac{1}{3ax}\left(2e^{ax^{2}}+ax^{2}-2% \right),

a,x>0

.

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and $x$ : real variable
Referenced by:: §7.8
Permalink:: http://dlmf.nist.gov/7.8.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §7.8 and Ch.7

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7.8.7

\frac{\sinh x^{2}}{x}<{\mathrm{e}}^{x^{2}}F\left(x\right)=\int_{0}^{x}{\mathrm% {e}}^{t^{2}}\,\mathrm{d}t<\frac{{\mathrm{e}}^{x^{2}}-1}{x},

x>0

.

ⓘ

Symbols:: $F\left(\NVar{z}\right)$ : Dawson’s integral, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral and $x$ : real variable
Referenced by:: Erratum (V1.1.5) for Additions
Permalink:: http://dlmf.nist.gov/7.8.E7
Encodings:: pMML, png, TeX
Addition (effective with 1.1.5):: A new inequality was added.
See also:: Annotations for §7.8 and Ch.7

►The function

F\left(x\right)/\sqrt{1-{\mathrm{e}}^{-2x^{2}}}

is strictly decreasing for

x>0

. For these and similar results for Dawson’s integral

F\left(x\right)

see Janssen (2021). …

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•

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.

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•

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.

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	Open Source												With Book						Commercial
…
6 Exponential, Logarithmic, Sine, and Cosine Integrals
6.21(ii) $E_{1}\left(x\right)$ , $\operatorname{Ei}\left(x\right)$ , $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $\operatorname{Shi}\left(x\right)$ , $\operatorname{Chi}\left(x\right)$ , $x\in\mathbb{R}$	✓	✓	✓	✓	✓	✓	✓		✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓
6.21(iii) $E_{1}\left(z\right)$ , $\operatorname{Si}\left(z\right)$ , $\operatorname{Ci}\left(z\right)$ , $\operatorname{Shi}\left(z\right)$ , $\operatorname{Chi}\left(z\right)$ , $z\in\mathbb{C}$	✓	✓				✓			✓		✓	✓	✓							✓	✓	✓
…
7.25(ii) $\operatorname{erf}x$ , $\operatorname{erfc}x$ , $\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(x\right)$ , $x\in\mathbb{R}$	✓		✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓		✓	✓	✓	✓	✓	✓	NMS
…
8.28(vii) $E_{p}\left(z\right)$ , $z,p\in\mathbb{C}$	✓										✓	✓								✓	✓	✓
…

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… ►For values of

K,K^{\prime}

when

k^{2}=\frac{1}{2}

(lemniscatic case) see §23.5(iii), and for

k^{2}=e^{\mathrm{i}\pi/3}

(equianharmonic case) see §23.5(v). …

§7.4 Symmetry

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7.4.2

\operatorname{erfc}\left(-z\right)=2-\operatorname{erfc}\left(z\right),

ⓘ

Symbols:: $\operatorname{erfc}\NVar{z}$ : complementary error function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/7.4.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §7.4 and Ch.7

►

7.4.3

w\left(-z\right)=2e^{-z^{2}}-w\left(z\right).

ⓘ

Symbols:: $w\left(\NVar{z}\right)$ : Faddeeva (or Faddeyeva) function, $\mathrm{e}$ : base of natural logarithm and $z$ : complex variable
A&S Ref:: 7.1.11
Permalink:: http://dlmf.nist.gov/7.4.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §7.4 and Ch.7

►

7.4.4

F\left(-z\right)=-F\left(z\right).

ⓘ

Symbols:: $F\left(\NVar{z}\right)$ : Dawson’s integral and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/7.4.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §7.4 and Ch.7

►

C\left(-z\right)=-C\left(z\right),

…

complementary exponential integral

21—30 of 48 matching pages

21: 8.4 Special Values

22: 7.2 Definitions

23: 12.7 Relations to Other Functions

24: 3.5 Quadrature

25: 7.1 Special Notation

26: 7.8 Inequalities

§7.8 Inequalities

27: 7.23 Tables

28: Software Index

29: 22.5 Special Values

30: 7.4 Symmetry

§7.4 Symmetry