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1: 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.
  • 2: 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
    3: Bibliography
  • M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.
  • A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.
  • N. I. Akhiezer (1988) Lectures on Integral Transforms. Translations of Mathematical Monographs, Vol. 70, American Mathematical Society, Providence, RI.
  • S. V. Aksenov, M. A. Savageau, U. D. Jentschura, J. Becher, G. Soff, and P. J. Mohr (2003) Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics. Comput. Phys. Comm. 150 (1), pp. 1–20.
  • D. E. Amos (1989) Repeated integrals and derivatives of K Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.
  • 4: 19.36 Methods of Computation
    §19.36(ii) Quadratic Transformations
    The incomplete integrals R F ( x , y , z ) and R G ( x , y , z ) can be computed by successive transformations in which two of the three variables converge quadratically to a common value and the integrals reduce to R C , accompanied by two quadratically convergent series in the case of R G ; compare Carlson (1965, §§5,6). … Computation of Legendre’s integrals of all three kinds by quadratic transformation is described by Cazenave (1969, pp. 128–159, 208–230). Quadratic transformations can be applied to compute Bulirsch’s integrals (§19.2(iii)). … For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). …
    5: Bibliography K
  • R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.
  • M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
  • A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.
  • T. H. Koornwinder (2009) The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.
  • V. I. Krylov and N. S. Skoblya (1985) A Handbook of Methods of Approximate Fourier Transformation and Inversion of the Laplace Transformation. Mir, Moscow.
  • 6: Bibliography C
  • R. G. Campos (1995) A quadrature formula for the Hankel transform. Numer. Algorithms 9 (2), pp. 343–354.
  • R. Chelluri, L. B. Richmond, and N. M. Temme (2000) Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.
  • N. B. Christensen (1990) Optimized fast Hankel transform filters. Geophysical Prospecting 38 (5), pp. 545–568.
  • M. Colman, A. Cuyt, and J. Van Deun (2011) Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.
  • M. D. Cooper, R. H. Jeppesen, and M. B. Johnson (1979) Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.
  • 7: Bibliography D
  • B. Davies (1984) Integral Transforms and their Applications. 2nd edition, Applied Mathematical Sciences, Vol. 25, Springer-Verlag, New York.
  • L. Debnath and D. Bhatta (2015) Integral transforms and their applications. Third edition, CRC Press, Boca Raton, FL.
  • G. Doetsch (1955) Handbuch der Laplace-Transformation. Bd. II. Anwendungen der Laplace-Transformation. 1. Abteilung. Birkhäuser Verlag, Basel und Stuttgart (German).
  • B. Döring (1966) Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.
  • T. M. Dunster (1989) Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.
  • 8: 20.11 Generalizations and Analogs
    If both m , n are positive, then G ( m , n ) allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)): … In the case z = 0 identities for theta functions become identities in the complex variable q , with | q | < 1 , that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7). … However, in this case q is no longer regarded as an independent complex variable within the unit circle, because k is related to the variable τ = τ ( k ) of the theta functions via (20.9.2). …
    9: Bibliography S
  • K. L. Sala (1989) Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.
  • J. L. Schiff (1999) The Laplace Transform: Theory and Applications. Undergraduate Texts in Mathematics, Springer-Verlag, New York.
  • O. A. Sharafeddin, H. F. Bowen, D. J. Kouri, and D. K. Hoffman (1992) Numerical evaluation of spherical Bessel transforms via fast Fourier transforms. J. Comput. Phys. 100 (2), pp. 294–296.
  • A. Sharples (1967) Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
  • J. R. Stembridge (1995) A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.
  • 10: Bibliography G
  • W. Gautschi (1994) Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.
  • A. G. Gibbs (1973) Problem 72-21, Laplace transforms of Airy functions. SIAM Rev. 15 (4), pp. 796–798.
  • A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
  • Ya. I. Granovskiĭ, I. M. Lutzenko, and A. S. Zhedanov (1992) Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.
  • W. Groenevelt (2007) Fourier transforms related to a root system of rank 1. Transform. Groups 12 (1), pp. 77–116.