Mellin transform methods
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1: 2.5 Mellin Transform Methods
§2.5 Mellin Transform Methods
… ►§2.5(ii) Extensions
… ►To apply the Mellin transform method outlined in §2.5(i), we require the transforms and to have a common strip of analyticity. … ►The Mellin transform method can also be extended to derive asymptotic expansions of multidimensional integrals having algebraic or logarithmic singularities, or both; see Wong (1989, Chapter 3), Paris and Kaminski (2001, Chapter 7), and McClure and Wong (1987). See also Brüning (1984) for a different approach. …2: 2.3 Integrals of a Real Variable
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►where is the Mellin transform of (§2.5(i)).
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§2.3(iii) Laplace’s Method
… ►The integral (2.3.24) transforms into … ►§2.3(vi) Asymptotics of Mellin Transforms
►For the asymptotics of the Mellin transform as see Frenzen (1987b), Sidi (1985, 2011).3: 3.5 Quadrature
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►Stroud and Secrest (1966) includes computational methods and extensive tables.
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►Oscillatory integral transforms are treated in Wong (1982) by a method based on Gaussian quadrature.
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►Further methods are given in Clendenin (1966) and Lyness (1985).
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►Other contour integrals occur in standard integral transforms or their inverses, for example, Hankel transforms (§10.22(v)), Kontorovich–Lebedev transforms (§10.43(v)), and Mellin transforms (§1.14(iv)).
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4: Bibliography O
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Tables of Fourier Transforms and Fourier Transforms of Distributions.
Springer-Verlag, Berlin.
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Tables of Mellin Transforms.
Springer-Verlag, Berlin-New York.
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Asymptotic Enumeration Methods.
In Handbook of Combinatorics, Vol. 2, L. Lovász, R. L. Graham, and M. Grötschel (Eds.),
pp. 1063–1229.
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Computing : The combinatorial method.
Revista do DETUA 4 (6), pp. 759–768.
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An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series.
J. Inst. Math. Appl. 20 (3), pp. 379–391.
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5: 2.6 Distributional Methods
§2.6 Distributional Methods
… ►§2.6(ii) Stieltjes Transform
… ► being the Mellin transform of or its analytic continuation (§2.5(ii)). … ►For a more detailed discussion of the derivation of asymptotic expansions of Stieltjes transforms by the distribution method, see McClure and Wong (1978) and Wong (1989, Chapter 6). … ►where is the Mellin transform of or its analytic continuation. …6: Bibliography S
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A global Newton method for the zeros of cylinder functions.
Numer. Algorithms 18 (3-4), pp. 259–276.
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Gaussian-transform method for molecular integrals. I. Formulation for energy integrals.
J. Chem. Phys. 43 (2), pp. 398–414.
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Asymptotic expansions of Mellin transforms and analogues of Watson’s lemma.
SIAM J. Math. Anal. 16 (4), pp. 896–906.
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Practical Extrapolation Methods: Theory and Applications.
Cambridge Monographs on Applied and Computational Mathematics, Vol. 10, Cambridge University Press, Cambridge.
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Asymptotic expansion of Mellin transforms in the complex plane.
Int. J. Pure Appl. Math. 71 (3), pp. 465–480.
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7: Bibliography W
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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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Wave functions for large arguments by the amplitude-phase method.
Phys. Rev. 52, pp. 1123–1127.
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The method of Darboux.
J. Approximation Theory 10 (2), pp. 159–171.
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On a uniform treatment of Darboux’s method.
Constr. Approx. 21 (2), pp. 225–255.
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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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8: Bibliography F
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Un théorème de Paley-Wiener pour la transformation de Fourier sur un espace riemannien symétrique de rang un.
J. Funct. Anal. 49 (2), pp. 230–268.
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Calculation of elliptic integrals of the third kind by means of Gauss’ transformation.
Math. Comp. 19 (89), pp. 97–104.
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Algebraic methods to compute Mathieu functions.
J. Phys. A 34 (17), pp. 3541–3551.
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On the asymptotic expansion of Mellin transforms.
SIAM J. Math. Anal. 18 (1), pp. 273–282.
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Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions.
In Progress in Optics, E. Wolf (Ed.),
Vol. 9, pp. 311–407.
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9: Bibliography P
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Asymptotics and Mellin-Barnes Integrals.
Cambridge University Press, Cambridge.
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Smoothing of the Stokes phenomenon using Mellin-Barnes integrals.
J. Comput. Appl. Math. 41 (1-2), pp. 117–133.
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Exactification of the method of steepest descents: The Bessel functions of large order and argument.
Proc. Roy. Soc. London Ser. A 460, pp. 2737–2759.
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Modified Clenshaw-Curtis method for the computation of Bessel function integrals.
BIT 23 (3), pp. 370–381.
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A method for computing Bessel function integrals.
J. Comput. Phys. 75 (2), pp. 334–344.