relations to Dawson integral and exponential integral

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1: 7.5 Interrelations

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2: 12.7 Relations to Other Functions

§12.7 Relations to Other Functions

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§12.7(i) Hermite Polynomials

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§12.7(ii) Error Functions, Dawson’s Integral, and Probability Function

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§12.7(iii) Modified Bessel Functions

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§12.7(iv) Confluent Hypergeometric Functions

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3: Bibliography C

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Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric $R$-functions. Math. Comp. 75 (255), pp. 1309–1318.

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External Links:

ISSN 0025-5718, Document, MathReview, ZentralBlatt

Referenced by:

§19.25(v), §19.29(i), §22.14(ii).

Encodings:

BibTeX

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A numerical method for generalized exponential integrals. Comput. Math. Appl. 14 (4), pp. 261–268.

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External Links:

ISSN 0898-1221, MathReview (R. P. Boas), ZentralBlatt

Referenced by:

§8.25(v).

Encodings:

BibTeX

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Chebyshev approximations for Dawson’s integral. Math. Comp. 24 (109), pp. 171–178.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

4th item.

Encodings:

BibTeX

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Performance evaluation of programs related to the real gamma function. ACM Trans. Math. Software 17 (1), pp. 46–54.

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External Links:

ISSN 0098-3500, Document, MathReview, ZentralBlatt

Referenced by:

§5.24(ii), §5.24(iii).

Encodings:

BibTeX

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Applications of the complex exponential integral. Math. Comp. 15 (73), pp. 1–6.

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External Links:

FullText, MathReview (J. C. P. Miller), ZentralBlatt

Referenced by:

§6.7(iv).

Encodings:

BibTeX

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4: 7.2 Definitions

… ► $\mathrm{erf}z$, $\mathrm{erfc}z$, and $w\left(z\right)$ are entire functions of $z$, as is $F\left(z\right)$ in the next subsection. … ►

§7.2(ii) Dawson’s Integral

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§7.2(iii) Fresnel Integrals

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§7.2(iv) Auxiliary Functions

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§7.2(v) Goodwin–Staton Integral

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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 $\mathrm{erf}z$; 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 $𝖵(x,t)$. ►Alternative notations are $Q(z)=\frac{1}{2}\mathrm{erfc}\left(z/\sqrt{2}\right)$, $P(z)=\mathrm{\Phi }(z)=\frac{1}{2}\mathrm{erfc}\left(-z/\sqrt{2}\right)$, $\mathrm{Erf}z=\frac{1}{2}\sqrt{\pi }\mathrm{erf}z$, $\mathrm{Erfi}z={\mathrm{e}}^{z^2}F\left(z\right)$, $C_1(z)=C\left(\sqrt{2/\pi }z\right)$, $S_1(z)=S\left(\sqrt{2/\pi }z\right)$, $C_2(z)=C\left(\sqrt{2z/\pi }\right)$, $S_2(z)=S\left(\sqrt{2z/\pi }\right)$. … ►

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 $e_n(nx)/{\mathrm{e}}^{nx}$ follows from (8.11.14) and (8.11.16). …