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21: 25.10 Zeros
Calculations relating to the zeros on the critical line make use of the real-valued function … Calculations based on the Riemann–Siegel formula reveal that the first ten billion zeros of ζ ( s ) in the critical strip are on the critical line (van de Lune et al. (1986)). …
22: 9.16 Physical Applications
The use of Airy function and related uniform asymptotic techniques to calculate amplitudes of polarized rainbows can be found in Nussenzveig (1992) and Adam (2002). …
23: Bibliography T
  • J. D. Talman (1983) LSFBTR: A subroutine for calculating spherical Bessel transforms. Comput. Phys. Comm. 30 (1), pp. 93–99.
  • W. J. Thompson (1994) Angular Momentum: An Illustrated Guide to Rotational Symmetries for Physical Systems. A Wiley-Interscience Publication, John Wiley & Sons Inc., New York.
  • R. F. Tooper and J. Mark (1968) Simplified calculation of Ei ( x ) for positive arguments, and a short table of Shi ( x ) . Math. Comp. 22 (102), pp. 448–449.
  • 24: Bibliography
  • R. M. Aarts and A. J. E. M. Janssen (2016) Efficient approximation of the Struve functions 𝐇 n occurring in the calculation of sound radiation quantities. The Journal of the Acoustical Society of America 140 (6), pp. 4154–4160.
  • R. W. Abernathy and R. P. Smith (1993) Algorithm 724: Program to calculate F-percentiles. ACM Trans. Math. Software 19 (4), pp. 481–483.
  • G. Allasia and R. Besenghi (1987b) Numerical calculation of incomplete gamma functions by the trapezoidal rule. Numer. Math. 50 (4), pp. 419–428.
  • G. Allasia and R. Besenghi (1989) Numerical Calculation of the Riemann Zeta Function and Generalizations by Means of the Trapezoidal Rule. In Numerical and Applied Mathematics, Part II (Paris, 1988), C. Brezinski (Ed.), IMACS Ann. Comput. Appl. Math., Vol. 1, pp. 467–472.
  • 25: Bibliography L
  • W. J. Lentz (1976) Generating Bessel functions in Mie scattering calculations using continued fractions. Applied Optics 15 (3), pp. 668–671.
  • E. Lindelöf (1905) Le Calcul des Résidus et ses Applications à la Théorie des Fonctions. Gauthier-Villars, Paris (French).
  • É. Lucas (1891) Théorie des nombres. Tome I: Le calcul des nombres entiers, le calcul des nombres rationnels, la divisibilité arithmétique. Gauthier-Villars, Paris (French).
  • 26: 9.19 Approximations
  • Moshier (1989, §6.14) provides minimax rational approximations for calculating Ai ( x ) , Ai ( x ) , Bi ( x ) , Bi ( x ) . They are in terms of the variable ζ , where ζ = 2 3 x 3 / 2 when x is positive, ζ = 2 3 ( x ) 3 / 2 when x is negative, and ζ = 0 when x = 0 . The approximations apply when 2 ζ < , that is, when 3 2 / 3 x < or < x 3 2 / 3 . The precision in the coefficients is 21S.

  • 27: 33.23 Methods of Computation
    Bardin et al. (1972) describes ten different methods for the calculation of F and G , valid in different regions of the ( η , ρ )-plane. …
    28: Bibliography P
  • B. Pichon (1989) Numerical calculation of the generalized Fermi-Dirac integrals. Comput. Phys. Comm. 55 (2), pp. 127–136.
  • A. Poquérusse and S. Alexiou (1999) Fast analytic formulas for the modified Bessel functions of imaginary order for spectral line broadening calculations. J. Quantit. Spec. and Rad. Trans. 62 (4), pp. 389–395.
  • 29: 23.22 Methods of Computation
    §23.22(ii) Lattice Calculations
    The corresponding values of e 1 , e 2 , e 3 are calculated from (23.6.2)–(23.6.4), then g 2 and g 3 are obtained from (23.3.6) and (23.3.7). …
    30: Bibliography C
  • M. Chellali (1988) Accélération de calcul de nombres de Bernoulli. J. Number Theory 28 (3), pp. 347–362 (French).
  • W. W. Clendenin (1966) A method for numerical calculation of Fourier integrals. Numer. Math. 8 (5), pp. 422–436.
  • J. A. Cochran (1963) Further formulas for calculating approximate values of the zeros of certain combinations of Bessel functions. IEEE Trans. Microwave Theory Tech. 11 (6), pp. 546–547.
  • J. N. L. Connor and D. Farrelly (1981) Molecular collisions and cusp catastrophes: Three methods for the calculation of Pearcey’s integral and its derivatives. Chem. Phys. Lett. 81 (2), pp. 306–310.
  • J. N. L. Connor and D. C. Mackay (1979) Calculation of angular distributions in complex angular momentum theories of elastic scattering. Molecular Physics 37 (6), pp. 1703–1712.