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1: 1.14 Integral Transforms
§1.14 Integral Transforms
§1.14(i) Fourier Transform
§1.14(iii) Laplace Transform
Fourier Transform
Laplace Transform
2: 2.7 Differential Equations
To include the point at infinity in the foregoing classification scheme, we transform it into the origin by replacing z in (2.7.1) with 1 / z ; see Olver (1997b, pp. 153–154). … See §2.11(v) for other examples. The exceptional case f 0 2 = 4 g 0 is handled by Fabry’s transformation: …The transformed differential equation either has a regular singularity at t = , or its characteristic equation has unequal roots. …
3: 20 Theta Functions
Chapter 20 Theta Functions
4: 20.10 Integrals
§20.10(i) Mellin Transforms with respect to the Lattice Parameter
20.10.1 0 x s 1 θ 2 ( 0 | i x 2 ) d x = 2 s ( 1 2 s ) π s / 2 Γ ( 1 2 s ) ζ ( s ) , s > 1 ,
20.10.2 0 x s 1 ( θ 3 ( 0 | i x 2 ) 1 ) d x = π s / 2 Γ ( 1 2 s ) ζ ( s ) , s > 1 ,
20.10.3 0 x s 1 ( 1 θ 4 ( 0 | i x 2 ) ) d x = ( 1 2 1 s ) π s / 2 Γ ( 1 2 s ) ζ ( s ) , s > 0 .
§20.10(ii) Laplace Transforms with respect to the Lattice Parameter
5: Bibliography N
  • D. Naylor (1989) On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.
  • D. Naylor (1990) On an asymptotic expansion of the Kontorovich-Lebedev transform. Applicable Anal. 39 (4), pp. 249–263.
  • D. Naylor (1996) On an asymptotic expansion of the Kontorovich-Lebedev transform. Methods Appl. Anal. 3 (1), pp. 98–108.
  • W. J. Nellis and B. C. Carlson (1966) Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.
  • E. W. Ng and M. Geller (1969) A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.
  • 6: Bibliography F
  • FDLIBM (free C library)
  • S. Fempl (1960) Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).
  • H. E. Fettis and J. C. Caslin (1964) Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.
  • C. L. Frenzen (1987b) On the asymptotic expansion of Mellin transforms. SIAM J. Math. Anal. 18 (1), pp. 273–282.
  • G. Freud (1969) On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.
  • 7: Bibliography W
  • R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.
  • G. N. Watson (1910) The cubic transformation of the hypergeometric function. Quart. J. Pure and Applied Math. 41, pp. 70–79.
  • F. J. W. Whipple (1927) Some transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 26 (2), pp. 257–272.
  • D. V. Widder (1979) The Airy transform. Amer. Math. Monthly 86 (4), pp. 271–277.
  • D. V. Widder (1941) The Laplace Transform. Princeton Mathematical Series, v. 6, Princeton University Press, Princeton, NJ.
  • 8: 20.7 Identities
    §20.7(iv) Reduction Formulas for Products
    §20.7(vi) Landen Transformations
    See Lawden (1989, pp. 19–20). …
    §20.7(viii) Transformations of Lattice Parameter
    §20.7(ix) Addendum to 20.7(iv) Reduction Formulas for Products
    9: 15.14 Integrals
    §15.14 Integrals
    The Mellin transform of the hypergeometric function of negative argument is given by … Fourier transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §§1.14 and 2.14). 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). …Hankel transforms of hypergeometric functions are given in Oberhettinger (1972, §1.17) and Erdélyi et al. (1954b, §8.17). …
    10: 12.16 Mathematical Applications
    PCFs are also used in integral transforms with respect to the parameter, and inversion formulas exist for kernels containing PCFs. …Integral transforms and sampling expansions are considered in Jerri (1982).