Mellin transform with respect to lattice parameter
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11—20 of 910 matching pages
11: Bibliography W
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The analyticity of Jacobian functions with respect to the parameter
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Proc. Roy. Soc. London Ser A 459, pp. 2569–2574.
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The distribution of the zeros of Jacobian elliptic functions with respect to the parameter
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Comput. Methods Funct. Theory 9 (2), pp. 579–591.
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Extension of a quadratic transformation due to Whipple with an application.
Adv. Difference Equ., pp. 2013:157, 8.
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The cubic transformation of the hypergeometric function.
Quart. J. Pure and Applied Math. 41, pp. 70–79.
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Explicit error terms for asymptotic expansions of Mellin convolutions.
J. Math. Anal. Appl. 72 (2), pp. 740–756.
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12: 13.10 Integrals
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§13.10(ii) Laplace Transforms
… ►§13.10(iii) Mellin Transforms
… ►For additional Mellin transforms see Erdélyi et al. (1954a, §§6.9, 7.5), Marichev (1983, pp. 283–287), and Oberhettinger (1974, §§1.13, 2.8). ►§13.10(iv) Fourier Transforms
… ►§13.10(v) Hankel Transforms
…13: 19.13 Integrals of Elliptic Integrals
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§19.13(i) Integration with Respect to the Modulus
… ►§19.13(ii) Integration with Respect to the Amplitude
… ►Cvijović and Klinowski (1994) contains fractional integrals (with free parameters) for and , together with special cases. ►§19.13(iii) Laplace Transforms
►For direct and inverse Laplace transforms for the complete elliptic integrals , , and see Prudnikov et al. (1992a, §3.31) and Prudnikov et al. (1992b, §§3.29 and 4.3.33), respectively.14: 7.7 Integral Representations
15: 8.6 Integral Representations
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Mellin–Barnes Integrals
►In (8.6.10)–(8.6.12), is a real constant and the path of integration is indented (if necessary) so that in the case of (8.6.10) it separates the poles of the gamma function from the pole at , in the case of (8.6.11) it is to the right of all poles, and in the case of (8.6.12) it separates the poles of the gamma function from the poles at . ►
8.6.10
, ,
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8.6.11
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8.6.12
, .
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16: 9.10 Integrals
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§9.10(v) Laplace Transforms
… ►For Laplace transforms of products of Airy functions see Shawagfeh (1992). ►§9.10(vi) Mellin Transform
… ►§9.10(vii) Stieltjes Transforms
… ►§9.10(ix) Compendia
…17: 16.15 Integral Representations and Integrals
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16.15.1
, ,
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►For these and other formulas, including double Mellin–Barnes integrals, see Erdélyi et al. (1953a, §5.8).
These representations can be used to derive analytic continuations of the Appell functions, including convergent series expansions for large , large , or both.
For inverse Laplace transforms of Appell functions see Prudnikov et al. (1992b, §3.40).
18: 8.19 Generalized Exponential Integral
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§8.19(i) Definition and Integral Representations
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8.19.2
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►Integral representations of Mellin–Barnes type for follow immediately from (8.6.11), (8.6.12), and (8.19.1).
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►In Figures 8.19.2–8.19.5, height corresponds to the absolute value of the function and color to the phase.
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§8.19(vi) Relation to Confluent Hypergeometric Function
…19: 18.17 Integrals
20: 5.19 Mathematical Applications
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►As shown in Temme (1996b, §3.4), the results given in §5.7(ii) can be used to sum infinite series of rational functions.
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5.19.3
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