Goodwin–Staton integral

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1: 7.25 Software

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§7.25(vi) $ℱ\left(x\right)$, $G\left(x\right)$, $𝖴(x,t)$, $𝖵(x,t)$, $x\in ℝ$

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§7.25(vii) $ℱ\left(z\right)$, $G\left(z\right)$, $z\in ℂ$

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2: 7.23 Tables

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Abramowitz and Stegun (1964, Table 27.6) includes the Goodwin–Staton integral $G\left(x\right)$, $x=1(.1)3(.5)8$, 4D; also $G\left(x\right)+\ln x$, $x=0(.05)1$, 4D.

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3: 7.2 Definitions

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§7.2(v) Goodwin–Staton Integral

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7.2.12 $$G\left(z\right)={\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t^2}}{t+z}dt,$$ .

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Defines:

$G\left(z\right)$: Goodwin–Staton integral

Symbols:

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

A&S Ref:

27.6 (in different notation)

Permalink:

http://dlmf.nist.gov/7.2.E12

Encodings:

pMML, png, TeX

See also:

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

4: 7.1 Special Notation

… ►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 $𝖵(x,t)$. …

5: 7.5 Interrelations

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7.5.13 $$G\left(x\right)=\sqrt{\pi }F\left(x\right)-\frac{1}{2}{\mathrm{e}}^{-x^2}\mathrm{Ei}\left(x^2\right),$$ $x>0$.

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Symbols:

$F\left(z\right)$: Dawson’s integral, $G\left(z\right)$: Goodwin–Staton integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{Ei}\left(x\right)$: exponential integral and $x$: real variable

Referenced by:

§7.5

Permalink:

http://dlmf.nist.gov/7.5.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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6: 7.22 Methods of Computation

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§7.22(ii) Goodwin–Staton Integral

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7: Software Index

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	Open Source													With Book						Commercial
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7.25(vi) $ℱ\left(x\right)$, $G\left(x\right)$, $𝖴(x,t)$, $𝖵(x,t)$, $x\in ℝ$			✓	✓	✓						✓			✓	✓	✓	✓			✓	✓	✓	✓
7.25(vii) $ℱ\left(z\right)$, $G\left(z\right)$, $z\in ℂ$																					✓	✓	✓
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8: 7.12 Asymptotic Expansions

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§7.12(iii) Goodwin–Staton Integral

►See Olver (1997b, p. 115) for an expansion of $G\left(z\right)$ with bounds for the remainder for real and complex values of $z$.

9: Bibliography G

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Table of ${\int }_0^{\mathrm{\infty }}\frac{e^{-u^2}}{u+x}𝑑u$ . Quart. J. Mech. Appl. Math. 1 (1), pp. 319–326.

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External Links:

Document, MathReview (J. G. van der Corput), ZentralBlatt

Referenced by:

§7.22(ii).

Encodings:

BibTeX

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