Goodwin–Staton integral

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2: 7.23 Tables
• Abramowitz and Stegun (1964, Table 27.6) includes the GoodwinStaton integral $G\left(x\right)$, $x=1(.1)3(.5)8$, 4D; also $G\left(x\right)+\ln x$, $x=0(.05)1$, 4D.

• 3: 7.2 Definitions
§7.2(v) Goodwin–StatonIntegral
7.2.12 $G\left(z\right)=\int_{0}^{\infty}\frac{e^{-t^{2}}}{t+z}\mathrm{d}t,$ $|\operatorname{ph}z|<\pi$.
4: 7.1 Special Notation
The main functions treated in this chapter are the error function $\operatorname{erf}z$; the complementary error functions $\operatorname{erfc}z$ and $w\left(z\right)$; Dawson’s integral $F\left(z\right)$; the Fresnel integrals $\mathcal{F}\left(z\right)$, $C\left(z\right)$, and $S\left(z\right)$; the GoodwinStaton integral $G\left(z\right)$; the repeated integrals of the complementary error function $\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)$; the Voigt functions $\mathsf{U}\left(x,t\right)$ and $\mathsf{V}\left(x,t\right)$. …
5: 7.5 Interrelations
7.5.13 $G\left(x\right)=\sqrt{\pi}F\left(x\right)-\tfrac{1}{2}e^{-x^{2}}\mathrm{Ei}% \left(x^{2}\right),$ $x>0$.
7: Software Index
 Open Source With Book Commercial … 7.25(vi) $\mathcal{F}\left(x\right)$, $G\left(x\right)$, $\mathsf{U}\left(x,t\right)$, $\mathsf{V}\left(x,t\right)$, $x\in\mathbb{R}$ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ 7.25(vii) $\mathcal{F}\left(z\right)$, $G\left(z\right)$, $z\in\mathbb{C}$ ✓ ✓ ✓ …
8: 7.12 Asymptotic Expansions
§7.12(iii) Goodwin–StatonIntegral
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
• E. T. Goodwin and J. Staton (1948) Table of $\int_{0}^{\infty}\frac{e^{-u^{2}}}{u+x}\,du$ . Quart. J. Mech. Appl. Math. 1 (1), pp. 319–326.