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1: Bibliography I
  • IEEE (2018) IEEE Standard for Interval Arithmetic: IEEE Std 1788.1-2017. The Institute of Electrical and Electronics Engineers, Inc..
  • M. Ikonomou, P. Köhler, and A. F. Jacob (1995) Computation of integrals over the half-line involving products of Bessel functions, with application to microwave transmission lines. Z. Angew. Math. Mech. 75 (12), pp. 917–926.
  • A. Iserles, S. P. Nørsett, and S. Olver (2006) Highly Oscillatory Quadrature: The Story So Far. In Numerical Mathematics and Advanced Applications, A. Bermudez de Castro and others (Eds.), pp. 97–118.
  • M. E. H. Ismail and D. R. Masson (1994) q -Hermite polynomials, biorthogonal rational functions, and q -beta integrals. Trans. Amer. Math. Soc. 346 (1), pp. 63–116.
  • K. Iwasaki, H. Kimura, S. Shimomura, and M. Yoshida (1991) From Gauss to Painlevé: A Modern Theory of Special Functions. Aspects of Mathematics E, Vol. 16, Friedr. Vieweg & Sohn, Braunschweig, Germany.
  • 2: Publications
  • B. V. Saunders and Q. Wang (2005) Boundary/Contour Fitted Grid Generation for Effective Visualizations in a Digital Library of Mathematical Functions, Proceedings of the 9th International Conference on Numerical Grid Generation in Computational Field Simulations, San Jose, June 11–18, 2005. pp. 61–71. PDF
  • Q. Wang and B. V. Saunders (2005) Web-Based 3D Visualization in a Digital Library of Mathematical Functions, Proceedings of the Web3D Symposium, Bangor, UK, March 29–April 1, 2005. PDF
  • A. Youssef (2007) Methods of Relevance Ranking and Hit-content Generation in Math Search, Proceedings of Mathematical Knowledge Management (MKM2007), RISC, Hagenberg, Austria, June 27–30, 2007. PDF
  • B. I. Schneider, B. R. Miller and B. V. Saunders (2018) NIST’s Digital Library of Mathematial Functions, Physics Today 71, 2, 48 (2018), pp. 48–53. PDF
  • 3: Bibliography K
  • E. Kamke (1977) Differentialgleichungen: Lösungsmethoden und Lösungen. Teil I. B. G. Teubner, Stuttgart (German).
  • Y. S. Kim, A. K. Rathie, and R. B. Paris (2013) An extension of Saalschütz’s summation theorem for the series F r + 2 r + 3 . Integral Transforms Spec. Funct. 24 (11), pp. 916–921.
  • C. Kittel (1996) Introduction to Solid State Physics. 7th Edition edition, John Wiley and Sons, New York.
  • T. H. Koornwinder and M. Mazzocco (2018) Dualities in the q -Askey scheme and degenerate DAHA. Stud. Appl. Math. 141 (4), pp. 424–473.
  • T. Koornwinder, A. Kostenko, and G. Teschl (2018) Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator. Adv. Math. 333, pp. 796–821.
  • 4: Bibliography C
  • R. G. Campos (1995) A quadrature formula for the Hankel transform. Numer. Algorithms 9 (2), pp. 343–354.
  • B. C. Carlson (1963) Lauricella’s hypergeometric function F D . J. Math. Anal. Appl. 7 (3), pp. 452–470.
  • B. C. Carlson (2011) Permutation symmetry for theta functions. J. Math. Anal. Appl. 378 (1), pp. 42–48.
  • N. B. Christensen (1990) Optimized fast Hankel transform filters. Geophysical Prospecting 38 (5), pp. 545–568.
  • P. A. Clarkson and K. Jordaan (2018) Properties of generalized Freud polynomials. J. Approx. Theory 225, pp. 148–175.
  • 5: 25.21 Software
    §25.21(vii) Fermi–Dirac and Bose–Einstein Integrals
    6: Bibliography R
  • M. Rahman (2001) The Associated Classical Orthogonal Polynomials. In Special Functions 2000: Current Perspective and Future Directions (Tempe, AZ), NATO Sci. Ser. II Math. Phys. Chem., Vol. 30, pp. 255–279.
  • J. T. Ratnanather, J. H. Kim, S. Zhang, A. M. J. Davis, and S. K. Lucas (2014) Algorithm 935: IIPBF, a MATLAB toolbox for infinite integral of products of two Bessel functions. ACM Trans. Math. Softw. 40 (2), pp. 14:1–14:12.
  • M. Reed and B. Simon (1980) Methods of Modern Mathematical Physics, Vol. 1, Functional Analysis. Elsevier, New York.
  • W. P. Reinhardt (2018) Universality properties of Gaussian quadrature, the derivative rule, and a novel approach to Stieltjes inversion.
  • M. Robnik (1980) An extremum property of the n -dimensional sphere. J. Phys. A 13 (10), pp. L349–L351.
  • 7: 34.9 Graphical Method
    For an account of this method see Brink and Satchler (1993, Chapter VII). For specific examples of the graphical method of representing sums involving the 3 j , 6 j , and 9 j symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).
    8: 8.28 Software
    §8.28(vii) Generalized Exponential Integral for Complex Argument and/or Parameter
    9: 33.23 Methods of Computation
    §33.8 supplies continued fractions for F / F and H ± / H ± . Combined with the Wronskians (33.2.12), the values of F , G , and their derivatives can be extracted. … Curtis (1964a, §10) describes the use of series, radial integration, and other methods to generate the tables listed in §33.24. …
    §33.23(vii) WKBJ Approximations
    Hull and Breit (1959) and Barnett (1981b) give WKBJ approximations for F 0 and G 0 in the region inside the turning point: ρ < ρ tp ( η , ) .
    10: 10.74 Methods of Computation
    In the case of the modified Bessel function K ν ( z ) see especially Temme (1975). … It should be noted, however, that there is a difficulty in evaluating the coefficients A k ( ζ ) , B k ( ζ ) , C k ( ζ ) , and D k ( ζ ) , from the explicit expressions (10.20.10)–(10.20.13) when z is close to 1 owing to severe cancellation. … Similarly, to maintain stability in the interval 0 < x < the integration direction has to be forwards in the case of I ν ( x ) and backwards in the case of K ν ( x ) , with initial values obtained in an analogous manner to those for J ν ( x ) and Y ν ( x ) . … Then J n ( x ) and Y n ( x ) can be generated by either forward or backward recurrence on n when n < x , but if n > x then to maintain stability J n ( x ) has to be generated by backward recurrence on n , and Y n ( x ) has to be generated by forward recurrence on n . …
    §10.74(vii) Integrals