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1: 8.19 Generalized Exponential Integral

§8.19 Generalized Exponential Integral

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§8.19(i) Definition and Integral Representations

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§8.19(v) Recurrence Relation and Derivatives

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§8.19(vi) Relation to Confluent Hypergeometric Function

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§8.19(x) Integrals

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2: 6.2 Definitions and Interrelations

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§6.2(i) Exponential and Logarithmic Integrals

… ► $\mathrm{Ein}\left(z\right)$ is sometimes called the complementary exponential integral. … ► … ►The logarithmic integral is defined by … ►

§6.2(ii) Sine and Cosine Integrals

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3: 1.14 Integral Transforms

§1.14 Integral Transforms

… ►where the last integral denotes the Cauchy principal value (1.4.25). … ►Sufficient conditions for the integral to converge are that $s$ is a positive real number, and $f(t)=O\left(t^{-\delta }\right)$ as $t\to \mathrm{\infty }$, where $\delta >0$. … ►

§1.14(viii) Compendia

►For more extensive tables of the integral transforms of this section and tables of other integral transforms, see Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (2000), Marichev (1983), Oberhettinger (1972, 1974, 1990), Oberhettinger and Badii (1973), Oberhettinger and Higgins (1961), Prudnikov et al. (1986a, b, 1990, 1992a, 1992b).

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: 8.21 Generalized Sine and Cosine Integrals

§8.21 Generalized Sine and Cosine Integrals

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§8.21(iii) Integral Representations

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§8.21(iv) Interrelations

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§8.21(v) Special Values

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6: 7.18 Repeated Integrals of the Complementary Error Function

§7.18 Repeated Integrals of the Complementary Error Function

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Hermite Polynomials

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Confluent Hypergeometric Functions

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Parabolic Cylinder Functions

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Probability Functions

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7: 19.16 Definitions

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§19.16(i) Symmetric Integrals

… ►which is homogeneous and of degree $-a$ in the $z$’s, and unchanged when the same permutation is applied to both sets of subscripts $1,\mathrm{\dots },n$. …The $R$-function is often used to make a unified statement of a property of several elliptic integrals. … … ►Each of the four complete integrals (19.16.20)–(19.16.23) can be integrated to recover the incomplete integral: …

8: 36.2 Catastrophes and Canonical Integrals

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Canonical Integrals

… ► ${\mathrm{\Psi }}_1$ is related to the Airy function (§9.2): … … ►

§36.2(iii) Symmetries

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§36.2(iv) Addendum to 36.2(ii) Special Cases

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9: 19.2 Definitions

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§19.2(i) General Elliptic Integrals

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§19.2(ii) Legendre’s Integrals

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§19.2(iii) Bulirsch’s Integrals

… ►Lastly, corresponding to Legendre’s incomplete integral of the third kind we have … ►

§19.2(iv) A Related Function: $R_C(x,y)$

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10: 8 Incomplete Gamma and Related
Functions

Chapter 8 Incomplete Gamma and Related Functions

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