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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: Bibliography M
  • A. J. MacLeod (1996b) Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.
  • Fr. Mechel (1966) Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.
  • R. Metzler, J. Klafter, and J. Jortner (1999) Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems. Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.
  • D. S. Moak (1981) The q -analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.
  • L. Moser and M. Wyman (1958a) Asymptotic development of the Stirling numbers of the first kind. J. London Math. Soc. 33, pp. 133–146.
  • 3: 17.16 Mathematical Applications
    More recent applications are given in Gasper and Rahman (2004, Chapter 8) and Fine (1988, Chapters 1 and 2).
    4: 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.
  • 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.
  • J. F. Nye (1999) Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations. Institute of Physics Publishing, Bristol.
  • 5: 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.
  • N. J. Fine (1988) Basic Hypergeometric Series and Applications. Mathematical Surveys and Monographs, Vol. 27, American Mathematical Society, Providence, RI.
  • G. Freud (1969) On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.
  • 6: 28 Mathieu Functions and Hill’s Equation
    Chapter 28 Mathieu Functions and Hill’s Equation
    7: 33.22 Particle Scattering and Atomic and Molecular Spectra
    §33.22(i) Schrödinger Equation
    With e denoting here the elementary charge, the Coulomb potential between two point particles with charges Z 1 e , Z 2 e and masses m 1 , m 2 separated by a distance s is V ( s ) = Z 1 Z 2 e 2 / ( 4 π ε 0 s ) = Z 1 Z 2 α c / s , where Z j are atomic numbers, ε 0 is the electric constant, α is the fine structure constant, and is the reduced Planck’s constant. … For Z 1 Z 2 = 1 and m = m e , the electron mass, the scaling factors in (33.22.5) reduce to the Bohr radius, a 0 = / ( m e c α ) , and to a multiple of the Rydberg constant, R = m e c α 2 / ( 2 ) . … For bound-state problems only the exponentially decaying solution is required, usually taken to be the Whittaker function W η , + 1 2 ( 2 ρ ) . …
    8: 20 Theta Functions
    Chapter 20 Theta Functions
    9: 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.
  • C. Krattenthaler (1993) HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively q -binomial sums and basic hypergeometric series. Séminaire Lotharingien de Combinatoire 30, pp. 61–76.
  • 10: 17 q-Hypergeometric and Related Functions