Dawson integral

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11: 7.14 Integrals

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7.14.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{2\mathrm{i}at}\mathrm{erfc}\left(bt\right)dt=\frac{1}{a\sqrt{\pi }}F\left(\frac{a}{b}\right)+\frac{\mathrm{i}}{2a}\left(1-{\mathrm{e}}^{-{(a/b)}^2}\right),$$ $a\in ℂ$, .

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Symbols:

$F\left(z\right)$: Dawson’s integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{ph}$: phase

A&S Ref:

7.4.18 (in different form)

Referenced by:

§7.14(i)

Permalink:

http://dlmf.nist.gov/7.14.E1

Encodings:

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See also:

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

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12: 8.11 Asymptotic Approximations and Expansions

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8.11.11 $${\gamma }^{\ast }(1-a,-x)=x^{a-1}\left(-\cos \left(\pi a\right)+\frac{\sin \left(\pi a\right)}{\pi }\left(2\sqrt{\pi }F\left(y\right)+\frac{2}{3}\sqrt{\frac{2\pi }{a}}\left(1-y^2\right)\right){\mathrm{e}}^{y^2}+O\left(a^{-1}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $F\left(z\right)$: Dawson’s integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $\sin z$: sine function, $x$: real variable, $a$: parameter and $y$: change of variable

Permalink:

http://dlmf.nist.gov/8.11.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(iv), §8.11 and Ch.8

… ►For Dawson’s integral $F\left(y\right)$ see §7.2(ii). …

13: 8.12 Uniform Asymptotic Expansions for Large Parameter

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8.12.6 $$z^{-a}{\gamma }^{\ast }(-a,-z)=\cos \left(\pi a\right)-2\sin \left(\pi a\right)\left(\frac{{\mathrm{e}}^{\frac{1}{2}a{\eta }^2}}{\sqrt{\pi }}F\left(\eta \sqrt{a/2}\right)+T(a,\eta )\right),$$

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Symbols:

$F\left(z\right)$: Dawson’s integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $\sin z$: sine function, $z$: complex variable, $a$: parameter, $\eta (\lambda )$ and $T(a,\eta )$: function

Permalink:

http://dlmf.nist.gov/8.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §8.12 and Ch.8

►where $F\left(x\right)$ is Dawson’s integral; see §7.2(ii). …

14: 7.7 Integral Representations

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

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7.7.3 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-a{t}^2+2\mathrm{i}zt}dt=\frac{1}{2}\sqrt{\frac{\pi }{a}}{\mathrm{e}}^{-z^2/a}+\frac{\mathrm{i}}{\sqrt{a}}F\left(\frac{z}{\sqrt{a}}\right),$$ $\mathrm{\Re }a>0$.

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Symbols:

$F\left(z\right)$: Dawson’s integral, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

7.4.6 7.4.7 (in different notation)

Referenced by:

§7.14(i), §7.7(i)

Permalink:

http://dlmf.nist.gov/7.7.E3

Encodings:

pMML, png, TeX

See also:

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

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15: 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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12.7.4 $$V(-\frac{1}{2},z)=(2/\sqrt{\pi }){\mathrm{e}}^{\frac{1}{4}{z}^2}F\left(z/\sqrt{2}\right),$$

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Symbols:

$F\left(z\right)$: Dawson’s integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $V(a,z)$: parabolic cylinder function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/12.7.E4

Encodings:

pMML, png, TeX

See also:

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

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16: 7 Error Functions, Dawson’s and Fresnel Integrals

Chapter 7 Error Functions, Dawson’s and Fresnel Integrals

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17: 7.21 Physical Applications

… ►Efficient algorithms for computing the Faddeeva (or Faddeyeva) function are discussed in Wells (1999), a paper frequently cited in the astrophysics literature. … ►Dawson’s integral appears in de-convolving even more complex motional effects; see Pratt (2007). …

18: 7.22 Methods of Computation

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§7.22(i) Main Functions

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19: Nico M. Temme

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7 Error Functions, Dawson’s and Fresnel Integrals

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20: Bibliography J

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Bounds on Dawson’s integral occurring in the analysis of a line distribution network for electric vehicles. Eurandom Preprint Series Technical Report 14, Eurandom, Eindhoven, The Netherlands.

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Notes:

ISSN 1389-2355

External Links:

FullText

Referenced by:

§7.8, §7.8.

Encodings:

BibTeX

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