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1: 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 $\delta >0$. … ►

Laplace Transform

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§1.14(viii) Compendia

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2: 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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Other Integral Representations

… ►Integral representations of Mellin–Barnes type for $E_p\left(z\right)$ follow immediately from (8.6.11), (8.6.12), and (8.19.1). … ►

§8.19(x) Integrals

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3: 6.2 Definitions and Interrelations

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§6.2(i) Exponential and Logarithmic Integrals

… ► … ►The logarithmic integral is defined by … ►

§6.2(ii) Sine and Cosine Integrals

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4: 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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5: 7.2 Definitions

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§7.2(ii) Dawson’s Integral

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§7.2(iii) Fresnel Integrals

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Values at Infinity

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§7.2(iv) Auxiliary Functions

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§7.2(v) Goodwin–Staton Integral

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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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§7.18(i) Definition

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§7.18(iii) Properties

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Hermite Polynomials

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7: 19.16 Definitions

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§19.16(i) Symmetric Integrals

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§19.16(ii) $R_{-a}(𝐛;𝐳)$

… ► … ►All other elliptic cases are integrals of the second kind. … ►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

§36.2 Catastrophes and Canonical Integrals

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§36.2(i) Definitions

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Canonical Integrals

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§36.2(iii) Symmetries

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9: 19.2 Definitions

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§19.2(i) General Elliptic Integrals

… ►is called an elliptic integral. … ►

§19.2(ii) Legendre’s Integrals

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§19.2(iii) Bulirsch’s Integrals

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

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10: 25.12 Polylogarithms

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Integral Representation

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§25.12(iii) Fermi–Dirac and Bose–Einstein Integrals

►The Fermi–Dirac and Bose–Einstein integrals are defined by … ►In terms of polylogarithms … ►For a uniform asymptotic approximation for $F_s(x)$ see Temme and Olde Daalhuis (1990).