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31: Bibliography H
  • P. I. Hadži (1973) The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).
  • N. Hale and A. Townsend (2016) A fast FFT-based discrete Legendre transform. IMA J. Numer. Anal. 36 (4), pp. 1670–1684.
  • R. A. Handelsman and J. S. Lew (1970) Asymptotic expansion of Laplace transforms near the origin. SIAM J. Math. Anal. 1 (1), pp. 118–130.
  • E. W. Hansen (1985) Fast Hankel transform algorithm. IEEE Trans. Acoust. Speech Signal Process. 32 (3), pp. 666–671.
  • P. Henrici (1986) Applied and Computational Complex Analysis. Vol. 3: Discrete Fourier Analysis—Cauchy Integrals—Construction of Conformal Maps—Univalent Functions. Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons Inc.], New York.
  • 32: Bibliography T
  • J. D. Talman (1983) LSFBTR: A subroutine for calculating spherical Bessel transforms. Comput. Phys. Comm. 30 (1), pp. 93–99.
  • I. C. Tang (1969) Some definite integrals and Fourier series for Jacobian elliptic functions. Z. Angew. Math. Mech. 49, pp. 95–96.
  • A. Terras (1999) Fourier Analysis on Finite Groups and Applications. London Mathematical Society Student Texts, Vol. 43, Cambridge University Press, Cambridge.
  • E. C. Titchmarsh (1986a) Introduction to the Theory of Fourier Integrals. Third edition, Chelsea Publishing Co., New York.
  • G. P. Tolstov (1962) Fourier Series. Prentice-Hall Inc., Englewood Cliffs, N.J..
  • 33: 20.14 Methods of Computation
    The Fourier series of §20.2(i) usually converge rapidly because of the factors q ( n + 1 2 ) 2 or q n 2 , and provide a convenient way of calculating values of θ j ( z | τ ) . … For values of | q | near 1 the transformations of §20.7(viii) can be used to replace τ with a value that has a larger imaginary part and hence a smaller value of | q | . …In theory, starting from any value of τ , a finite number of applications of the transformations τ τ + 1 and τ 1 / τ will result in a value of τ with τ 3 / 2 ; see §23.18. …
    34: Bibliography L
  • D. Lemoine (1997) Optimal cylindrical and spherical Bessel transforms satisfying bound state boundary conditions. Comput. Phys. Comm. 99 (2-3), pp. 297–306.
  • M. J. Lighthill (1958) An Introduction to Fourier Analysis and Generalised Functions. Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, New York.
  • J. L. López, P. Pagola, and E. Pérez Sinusía (2013a) Asymptotics of the first Appell function F 1 with large parameters II. Integral Transforms Spec. Funct. 24 (12), pp. 982–999.
  • J. Lund (1985) Bessel transforms and rational extrapolation. Numer. Math. 47 (1), pp. 1–14.
  • J. N. Lyness (1971) Adjusted forms of the Fourier coefficient asymptotic expansion and applications in numerical quadrature. Math. Comp. 25 (113), pp. 87–104.
  • 35: 1.17 Integral and Series Representations of the Dirac Delta
    §1.17(ii) Integral Representations
    Formal interchange of the order of integration in the Fourier integral formula ((1.14.1) and (1.14.4)): …
    1.17.12 δ ( x a ) = 1 2 π e i ( x a ) t d t .
    Integral representation (1.17.12_1), (1.17.12_2) is the equivalent of the transform pairs, (1.14.9) & (1.14.11), (1.14.10) & (1.14.12), respectively. … Formal interchange of the order of summation and integration in the Fourier summation formula ((1.8.3) and (1.8.4)): …
    36: 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.
  • B. Deconinck and J. N. Kutz (2006) Computing spectra of linear operators using the Floquet-Fourier-Hill method. J. Comput. Phys. 219 (1), pp. 296–321.
  • H. Delange (1991) Sur les zéros réels des polynômes de Bernoulli. Ann. Inst. Fourier (Grenoble) 41 (2), pp. 267–309 (French).
  • G. Doetsch (1955) Handbuch der Laplace-Transformation. Bd. II. Anwendungen der Laplace-Transformation. 1. Abteilung. Birkhäuser Verlag, Basel und Stuttgart (German).
  • 37: 22.20 Methods of Computation
    §22.20(iii) Landen Transformations
    By application of the transformations given in §§22.7(i) and 22.7(ii), k or k can always be made sufficently small to enable the approximations given in §22.10(ii) to be applied. … am ( x , k ) can be computed from its definition (22.16.1) or from its Fourier series (22.16.9). …
    38: 18.2 General Orthogonal Polynomials
    §18.2(vii) Quadratic Transformations
    For such a system, functions f L w 2 ( ( a , b ) ) and sequences { λ n } ( n = 0 , 1 , 2 , ) satisfying n = 0 h n | λ n | 2 < can be related to each other in a similar way as was done for Fourier series in (1.8.1) and (1.8.2): …