# relations to Dawson integral and exponential integral

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##### 3: Bibliography C
• B. C. Carlson (2006b) Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric $R$-functions. Math. Comp. 75 (255), pp. 1309–1318.
• C. Chiccoli, S. Lorenzutta, and G. Maino (1987) A numerical method for generalized exponential integrals. Comput. Math. Appl. 14 (4), pp. 261–268.
• W. J. Cody, K. A. Paciorek, and H. C. Thacher (1970) Chebyshev approximations for Dawson’s integral. Math. Comp. 24 (109), pp. 171–178.
• W. J. Cody (1991) Performance evaluation of programs related to the real gamma function. ACM Trans. Math. Software 17 (1), pp. 46–54.
• M. S. Corrington (1961) Applications of the complex exponential integral. Math. Comp. 15 (73), pp. 1–6.
• ##### 4: 7.2 Definitions
$\operatorname{erf}z$, $\operatorname{erfc}z$, and $w\left(z\right)$ are entire functions of $z$, as is $F\left(z\right)$ in the next subsection. …
##### 5: 7.1 Special Notation
Unless otherwise noted, primes indicate derivatives with respect to the argument. 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 Goodwin–Staton 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 $\mathsf{V}\left(x,t\right)$. Alternative notations are $Q(z)=\tfrac{1}{2}\operatorname{erfc}\left(z/\sqrt{2}\right)$, $P(z)=\Phi(z)=\tfrac{1}{2}\operatorname{erfc}\left(-z/\sqrt{2}\right)$, $\operatorname{Erf}z=\tfrac{1}{2}\sqrt{\pi}\operatorname{erf}z$, $\operatorname{Erfi}z=e^{z^{2}}F\left(z\right)$, $C_{1}(z)=C\left(\sqrt{2/\pi}z\right)$, $S_{1}(z)=S\left(\sqrt{2/\pi}z\right)$, $C_{2}(z)=C\left(\sqrt{2z/\pi}\right)$, $S_{2}(z)=S\left(\sqrt{2z/\pi}\right)$. …
##### 6: 8.11 Asymptotic Approximations and Expansions
in both cases uniformly with respect to bounded real values of $y$. For Dawson’s integral $F\left(y\right)$ see §7.2(ii). …For related expansions involving Hermite polynomials see Pagurova (1965). … This reference also contains explicit formulas for the coefficients in terms of Stirling numbers. … With $x=1$, an asymptotic expansion of $e_{n}(nx)/e^{nx}$ follows from (8.11.14) and (8.11.16). …