Mellin%E2%80%93Barnes%20integrals

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1: 1.14 Integral Transforms

§1.14 Integral Transforms

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§1.14(iv) Mellin Transform

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Inversion

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Convolution

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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: 2.5 Mellin Transform Methods

§2.5 Mellin Transform Methods

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§2.5(ii) Extensions

… ►See also Brüning (1984) for a different approach. …

7: 15.14 Integrals

§15.14 Integrals

►The Mellin transform of the hypergeometric function of negative argument is given by … ►Mellin transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §6.9), Oberhettinger (1974, §1.15), and Marichev (1983, pp. 288–299). Inverse Mellin transforms are given in Erdélyi et al. (1954a, §7.5). … ►

8: 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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9: 19.16 Definitions

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§19.16(i) Symmetric Integrals

… ► … ►All other elliptic cases are integrals of the second kind. …(Note that $R_C(x,y)$ is not an elliptic integral.) … ►Each of the four complete integrals (19.16.20)–(19.16.23) can be integrated to recover the incomplete integral: …

10: 5.19 Mathematical Applications

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§5.19(ii) Mellin–Barnes Integrals

►Many special functions $f(z)$ can be represented as a Mellin–Barnes integral, that is, an integral of a product of gamma functions, reciprocals of gamma functions, and a power of $z$, the integration contour being doubly-infinite and eventually parallel to the imaginary axis at both ends. …By translating the contour parallel to itself and summing the residues of the integrand, asymptotic expansions of $f(z)$ for large $|z|$, or small $|z|$, can be obtained complete with an integral representation of the error term. …