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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 f ( 1 - z ) and h ( z ) 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
where f ( α ) is the Mellin transform of f ( t ) 2.5(i)). …
§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 f ( z ) = 0 t z - 1 f ( t ) d t as z see Frenzen (1987b), Sidi (1985, 2011).
3: 3.5 Quadrature
Stroud and Secrest (1966) includes computational methods and extensive tables. … Oscillatory integral transforms are treated in Wong (1982) by a method based on Gaussian quadrature. … Further methods are given in Clendenin (1966) and Lyness (1985). … Other contour integrals occur in standard integral transforms or their inverses, for example, Hankel transforms10.22(v)), Kontorovich–Lebedev transforms10.43(v)), and Mellin transforms1.14(iv)). …
4: Bibliography O
  • F. Oberhettinger (1990) Tables of Fourier Transforms and Fourier Transforms of Distributions. Springer-Verlag, Berlin.
  • F. Oberhettinger (1974) Tables of Mellin Transforms. Springer-Verlag, Berlin-New York.
  • A. M. Odlyzko (1995) Asymptotic Enumeration Methods. In Handbook of Combinatorics, Vol. 2, L. Lovász, R. L. Graham, and M. Grötschel (Eds.), pp. 1063–1229.
  • T. Oliveira e Silva (2006) Computing π ( x ) : The combinatorial method. Revista do DETUA 4 (6), pp. 759–768.
  • J. Oliver (1977) An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.
  • 5: 2.6 Distributional Methods
    §2.6 Distributional Methods
    §2.6(ii) Stieltjes Transform
    f ( z ) being the Mellin transform of f ( t ) 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 f ( z ) is the Mellin transform of f or its analytic continuation. …
    6: Bibliography S
  • J. Segura (1998) A global Newton method for the zeros of cylinder functions. Numer. Algorithms 18 (3-4), pp. 259–276.
  • I. Shavitt and M. Karplus (1965) Gaussian-transform method for molecular integrals. I. Formulation for energy integrals. J. Chem. Phys. 43 (2), pp. 398–414.
  • A. Sidi (1985) Asymptotic expansions of Mellin transforms and analogues of Watson’s lemma. SIAM J. Math. Anal. 16 (4), pp. 896–906.
  • A. Sidi (2003) Practical Extrapolation Methods: Theory and Applications. Cambridge Monographs on Applied and Computational Mathematics, Vol. 10, Cambridge University Press, Cambridge.
  • A. Sidi (2011) Asymptotic expansion of Mellin transforms in the complex plane. Int. J. Pure Appl. Math. 71 (3), pp. 465–480.
  • 7: Bibliography W
  • G. N. Watson (1910) The cubic transformation of the hypergeometric function. Quart. J. Pure and Applied Math. 41, pp. 70–79.
  • J. A. Wheeler (1937) Wave functions for large arguments by the amplitude-phase method. Phys. Rev. 52, pp. 1123–1127.
  • R. Wong and M. Wyman (1974) The method of Darboux. J. Approximation Theory 10 (2), pp. 159–171.
  • R. Wong and Y. Zhao (2005) On a uniform treatment of Darboux’s method. Constr. Approx. 21 (2), pp. 225–255.
  • R. Wong (1979) Explicit error terms for asymptotic expansions of Mellin convolutions. J. Math. Anal. Appl. 72 (2), pp. 740–756.
  • 8: Bibliography F
  • H. E. Fettis (1965) Calculation of elliptic integrals of the third kind by means of Gauss’ transformation. Math. Comp. 19 (89), pp. 97–104.
  • F. Feuillebois (1991) Numerical calculation of singular integrals related to Hankel transform. Comput. Math. Appl. 21 (2-3), pp. 87–94.
  • A. S. Fokas and M. J. Ablowitz (1982) On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23 (11), pp. 2033–2042.
  • C. L. Frenzen (1987b) On the asymptotic expansion of Mellin transforms. SIAM J. Math. Anal. 18 (1), pp. 273–282.
  • B. R. Frieden (1971) 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.
  • 9: Bibliography P
  • R. B. Paris and D. Kaminski (2001) Asymptotics and Mellin-Barnes Integrals. Cambridge University Press, Cambridge.
  • R. B. Paris (1992b) Smoothing of the Stokes phenomenon using Mellin-Barnes integrals. J. Comput. Appl. Math. 41 (1-2), pp. 117–133.
  • R. B. Paris (2004) 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.
  • R. Piessens and M. Branders (1983) Modified Clenshaw-Curtis method for the computation of Bessel function integrals. BIT 23 (3), pp. 370–381.
  • M. Puoskari (1988) A method for computing Bessel function integrals. J. Comput. Phys. 75 (2), pp. 334–344.