Mellin transform with respect to lattice parameter
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1: 1.14 Integral Transforms
§1.14 Integral Transforms
… ►§1.14(iv) Mellin Transform
… ►Inversion
… ►Parseval-type Formulas
… ►Convolution
…2: 2.5 Mellin Transform Methods
§2.5 Mellin Transform Methods
… ►The Mellin transform of is defined by …The inversion formula is given by … ► … ►To apply the Mellin transform method outlined in §2.5(i), we require the transforms and to have a common strip of analyticity. …3: 15.14 Integrals
§15.14 Integrals
►The Mellin transform of the hypergeometric function of negative argument is given by … ►Laplace transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §4.21), Oberhettinger and Badii (1973, §1.19), and Prudnikov et al. (1992a, §3.37). …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). …4: 20.10 Integrals
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§20.10(i) Mellin Transforms with respect to the Lattice Parameter
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20.10.1
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20.10.2
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§20.10(ii) Laplace Transforms with respect to the Lattice Parameter
… ►Then …5: 2.6 Distributional Methods
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§2.6(ii) Stieltjes Transform
… ► being the Mellin transform of or its analytic continuation (§2.5(ii)). … ►Corresponding results for the generalized Stieltjes transform …An application has been given by López (2000) to derive asymptotic expansions of standard symmetric elliptic integrals, complete with error bounds; see §19.27(vi). … ►where is the Mellin transform of or its analytic continuation. …6: Bibliography F
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Numerical calculation of singular integrals related to Hankel transform.
Comput. Math. Appl. 21 (2-3), pp. 87–94.
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Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. II.
J. Math. Anal. Appl. 7 (3), pp. 440–451.
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On a unified approach to transformations and elementary solutions of Painlevé equations.
J. Math. Phys. 23 (11), pp. 2033–2042.
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The transformation properties of the sixth Painlevé equation and one-parameter families of solutions.
Lett. Nuovo Cimento (2) 30 (17), pp. 539–544.
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On the asymptotic expansion of Mellin transforms.
SIAM J. Math. Anal. 18 (1), pp. 273–282.
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7: 13.23 Integrals
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§13.23(i) Laplace and Mellin Transforms
… ►For additional Laplace and Mellin transforms see Erdélyi et al. (1954a, §§4.22, 5.20, 6.9, 7.5), Marichev (1983, pp. 283–287), Oberhettinger and Badii (1973, §1.17), Oberhettinger (1974, §§1.13, 2.8), and Prudnikov et al. (1992a, §§3.34, 3.35). … ►§13.23(ii) Fourier Transforms
… ►§13.23(iii) Hankel Transforms
… ► …8: 2.3 Integrals of a Real Variable
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►Then
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►where is the Mellin transform of (§2.5(i)).
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►The integral (2.3.24) transforms into
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§2.3(vi) Asymptotics of Mellin Transforms
►For the asymptotics of the Mellin transform as see Frenzen (1987b), Sidi (1985, 2011).9: 11.7 Integrals and Sums
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11.7.10
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§11.7(iii) Laplace Transforms
►The following Laplace transforms of require for convergence, while those of require . … ►§11.7(iv) Integrals with Respect to Order
►For integrals of and with respect to the order , see Apelblat (1989). …10: 31.2 Differential Equations
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►All other homogeneous linear differential equations of the second order having four regular singularities in the extended complex plane, , can be transformed into (31.2.1).
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