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1: 12.16 Mathematical Applications
2: 13.23 Integrals
§13.23(iv) Integral Transforms in terms of Whittaker Functions
3: 13.10 Integrals
For integral transforms in terms of Whittaker functions see §13.23(iv). …
4: 19.15 Advantages of Symmetry
Symmetry in x , y , z of R F ( x , y , z ) , R G ( x , y , z ) , and R J ( x , y , z , p ) replaces the five transformations (19.7.2), (19.7.4)–(19.7.7) of Legendre’s integrals; compare (19.25.17). …(19.21.12) unifies the three transformations in §19.7(iii) that change the parameter of Legendre’s third integral. Symmetry allows the expansion (19.19.7) in a series of elementary symmetric functions that gives high precision with relatively few terms and provides the most efficient method of computing the incomplete integral of the third kind (§19.36(i)). … For the many properties of ellipses and triaxial ellipsoids that can be represented by elliptic integrals, any symmetry in the semiaxes remains obvious when symmetric integrals are used (see (19.30.5) and §19.33). …
5: Bibliography S
  • K. Soni (1980) Exact error terms in the asymptotic expansion of a class of integral transforms. I. Oscillatory kernels. SIAM J. Math. Anal. 11 (5), pp. 828–841.
  • 6: 19.36 Methods of Computation
    If α 2 = k 2 , then the method fails, but the function can be expressed by (19.6.13) in terms of E ( ϕ , k ) , for which Neuman (1969b) uses ascending Landen transformations. … Bulirsch (1969a, b) extend Bartky’s transformation to el3 ( x , k c , p ) by expressing it in terms of the first incomplete integral, a complete integral of the third kind, and a more complicated integral to which Bartky’s method can be applied. …
    7: 18.17 Integrals
    §18.17(v) Fourier Transforms
    §18.17(vi) Laplace Transforms
    8: 19.14 Reduction of General Elliptic Integrals
    (These four cases include 12 integrals in Abramowitz and Stegun (1964, p. 596).) … Legendre (1825–1832) showed that every elliptic integral can be expressed in terms of the three integrals in (19.1.2) supplemented by algebraic, logarithmic, and trigonometric functions. The classical method of reducing (19.2.3) to Legendre’s integrals is described in many places, especially Erdélyi et al. (1953b, §13.5), Abramowitz and Stegun (1964, Chapter 17), and Labahn and Mutrie (1997, §3). …A similar remark applies to the transformations given in Erdélyi et al. (1953b, §13.5) and to the choice among explicit reductions in the extensive table of Byrd and Friedman (1971), in which one limit of integration is assumed to be a branch point of the integrand at which the integral converges. …
    9: 2.11 Remainder Terms; Stokes Phenomenon
    Taking m = 10 in (2.11.2), the first three terms give us the approximation …The error term is, in fact, approximately 700 times the last term obtained in (2.11.4). … Two different asymptotic expansions in terms of elementary functions, (2.11.6) and (2.11.7), are available for the generalized exponential integral in the sector 1 2 π < ph z < 3 2 π . … For illustration, we give re-expansions of the remainder terms in the expansions (2.7.8) arising in differential-equation theory. … Subtraction of this result from the sum of the first 5 terms in (2.11.25) yields 0. …
    10: 10.22 Integrals
    §10.22 Integrals
    See also §1.17(ii) for an integral representation of the Dirac delta in terms of a product of Bessel functions. … Additional infinite integrals over the product of three Bessel functions (including modified Bessel functions) are given in Gervois and Navelet (1984, 1985a, 1985b, 1986a, 1986b).
    §10.22(v) Hankel Transform
    The Hankel transform (or Bessel transform) of a function f ( x ) is defined as …