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21: Bibliography S
  • R. Shail (1980) On integral representations for Lamé and other special functions. SIAM J. Math. Anal. 11 (4), pp. 702–723.
  • N. T. Shawagfeh (1992) The Laplace transforms of products of Airy functions. Dirāsāt Ser. B Pure Appl. Sci. 19 (2), pp. 7–11.
  • A. Sidi (2010) A simple approach to asymptotic expansions for Fourier integrals of singular functions. Appl. Math. Comput. 216 (11), pp. 3378–3385.
  • R. Sips (1965) Représentation asymptotique de la solution générale de l’équation de Mathieu-Hill. Acad. Roy. Belg. Bull. Cl. Sci. (5) 51 (11), pp. 1415–1446.
  • K. Soni (1980) Exact error terms in the asymptotic expansion of a class of integral transforms. I. Oscillatory kernels. SIAM J. Math. Anal. 11 (5), pp. 828–841.
  • 22: 10.75 Tables
  • Wills et al. (1982) tabulates j 0 , m , j 1 , m , y 0 , m , y 1 , m for m = 1 ( 1 ) 30 , 35D.

  • MacDonald (1989) tabulates the first 30 zeros, in ascending order of absolute value in the fourth quadrant, of the function J 0 ( z ) i J 1 ( z ) , 6D. (Other zeros of this function can be obtained by reflection in the imaginary axis).

  • Abramowitz and Stegun (1964, Chapter 11) tabulates 0 x J 0 ( t ) d t , 0 x Y 0 ( t ) d t , x = 0 ( .1 ) 10 , 10D; 0 x t 1 ( 1 J 0 ( t ) ) d t , x t 1 Y 0 ( t ) d t , x = 0 ( .1 ) 5 , 8D.

  • Abramowitz and Stegun (1964, Chapter 11) tabulates e x 0 x I 0 ( t ) d t , e x x K 0 ( t ) d t , x = 0 ( .1 ) 10 , 7D; e x 0 x t 1 ( I 0 ( t ) 1 ) d t , x e x x t 1 K 0 ( t ) d t , x = 0 ( .1 ) 5 , 6D.

  • Zhang and Jin (1996, pp. 296–305) tabulates 𝗃 n ( x ) , 𝗃 n ( x ) , 𝗒 n ( x ) , 𝗒 n ( x ) , 𝗂 n ( 1 ) ( x ) , 𝗂 n ( 1 ) ( x ) , 𝗄 n ( x ) , 𝗄 n ( x ) , n = 0 ( 1 ) 10 ( 10 ) 30 , 50, 100, x = 1 , 5, 10, 25, 50, 100, 8S; x 𝗃 n ( x ) , ( x 𝗃 n ( x ) ) , x 𝗒 n ( x ) , ( x 𝗒 n ( x ) ) (Riccati–Bessel functions and their derivatives), n = 0 ( 1 ) 10 ( 10 ) 30 , 50, 100, x = 1 , 5, 10, 25, 50, 100, 8S; real and imaginary parts of 𝗃 n ( z ) , 𝗃 n ( z ) , 𝗒 n ( z ) , 𝗒 n ( z ) , 𝗂 n ( 1 ) ( z ) , 𝗂 n ( 1 ) ( z ) , 𝗄 n ( z ) , 𝗄 n ( z ) , n = 0 ( 1 ) 15 , 20(10)50, 100, z = 4 + 2 i , 20 + 10 i , 8S. (For the notation replace j , y , i , k by 𝗃 , 𝗒 , 𝗂 ( 1 ) , 𝗄 , respectively.)

  • 23: Bibliography N
  • National Physical Laboratory (1961) Modern Computing Methods. 2nd edition, Notes on Applied Science, No. 16, Her Majesty’s Stationery Office, London.
  • G. Nemes (2017a) Error bounds for the asymptotic expansion of the Hurwitz zeta function. Proc. A. 473 (2203), pp. 20170363, 16.
  • E. Neuman (1969a) Elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 99–102.
  • E. Neuman (1969b) On the calculation of elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 91–94.
  • V. Yu. Novokshënov (1985) The asymptotic behavior of the general real solution of the third Painlevé equation. Dokl. Akad. Nauk SSSR 283 (5), pp. 1161–1165 (Russian).
  • 24: Bibliography L
  • D. Le (1985) An efficient derivative-free method for solving nonlinear equations. ACM Trans. Math. Software 11 (3), pp. 250–262.
  • A. Leitner and J. Meixner (1960) Eine Verallgemeinerung der Sphäroidfunktionen. Arch. Math. 11, pp. 29–39.
  • H. Lotsch and M. Gray (1964) Algorithm 244: Fresnel integrals. Comm. ACM 7 (11), pp. 660–661.
  • N. A. Lukaševič (1967b) On the theory of Painlevé’s third equation. Differ. Uravn. 3 (11), pp. 1913–1923 (Russian).
  • Y. L. Luke (1977a) Algorithms for rational approximations for a confluent hypergeometric function. Utilitas Math. 11, pp. 123–151.
  • 25: Bibliography C
  • L. Carlitz (1960) Note on Nörlund’s polynomial B n ( z ) . Proc. Amer. Math. Soc. 11 (3), pp. 452–455.
  • P. A. Clarkson (2003b) The fourth Painlevé equation and associated special polynomials. J. Math. Phys. 44 (11), pp. 5350–5374.
  • 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.
  • 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.
  • F. Cooper, A. Khare, and A. Saxena (2006) Exact elliptic compactons in generalized Korteweg-de Vries equations. Complexity 11 (6), pp. 30–34.
  • 26: Bibliography D
  • P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou (1999b) Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math. 52 (11), pp. 1335–1425.
  • S. C. Dhar (1940) Note on the addition theorem of parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 4, pp. 29–30.
  • E. Dorrer (1968) Algorithm 322. F-distribution. Comm. ACM 11 (2), pp. 116–117.
  • B. A. Dubrovin (1981) Theta functions and non-linear equations. Uspekhi Mat. Nauk 36 (2(218)), pp. 11–80 (Russian).
  • J. Dutka (1981) The incomplete beta function—a historical profile. Arch. Hist. Exact Sci. 24 (1), pp. 11–29.
  • 27: Bibliography K
  • G. A. Kalugin, D. J. Jeffrey, and R. M. Corless (2012) Bernstein, Pick, Poisson and related integral expressions for Lambert W . Integral Transforms Spec. Funct. 23 (11), pp. 817–829.
  • E. L. Kaplan (1948) Auxiliary table for the incomplete elliptic integrals. J. Math. Physics 27, pp. 11–36.
  • R. P. Kerr (1963) Gravitational field of a spinning mass as an example of algebraically special metrics. Phys. Rev. Lett. 11 (5), pp. 237–238.
  • K. S. Kölbig (1968) Algorithm 327: Dilogarithm [S22]. Comm. ACM 11 (4), pp. 270–271.
  • T. H. Koornwinder (2015) Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.
  • 28: Software Index
    29: 34.7 Basic Properties: 9 j Symbol
    34.7.1 { j 11 j 12 j 13 j 21 j 22 j 13 j 31 j 31 0 } = ( 1 ) j 12 + j 21 + j 13 + j 31 ( ( 2 j 13 + 1 ) ( 2 j 31 + 1 ) ) 1 2 { j 11 j 12 j 13 j 22 j 21 j 31 } .
    34.7.2 j 12 j 34 ( 2 j 12 + 1 ) ( 2 j 34 + 1 ) ( 2 j 13 + 1 ) ( 2 j 24 + 1 ) { j 1 j 2 j 12 j 3 j 4 j 34 j 13 j 24 j } { j 1 j 2 j 12 j 3 j 4 j 34 j 13 j 24 j } = δ j 13 , j 13 δ j 24 , j 24 .
    34.7.3 j 13 j 24 ( 1 ) 2 j 2 + j 24 + j 23 j 34 ( 2 j 13 + 1 ) ( 2 j 24 + 1 ) { j 1 j 2 j 12 j 3 j 4 j 34 j 13 j 24 j } { j 1 j 3 j 13 j 4 j 2 j 24 j 14 j 23 j } = { j 1 j 2 j 12 j 4 j 3 j 34 j 14 j 23 j } .
    34.7.4 ( j 13 j 23 j 33 m 13 m 23 m 33 ) { j 11 j 12 j 13 j 21 j 22 j 23 j 31 j 32 j 33 } = m r 1 , m r 2 , r = 1 , 2 , 3 ( j 11 j 12 j 13 m 11 m 12 m 13 ) ( j 21 j 22 j 23 m 21 m 22 m 23 ) ( j 31 j 32 j 33 m 31 m 32 m 33 ) ( j 11 j 21 j 31 m 11 m 21 m 31 ) ( j 12 j 22 j 32 m 12 m 22 m 32 ) .
    34.7.5 j ( 2 j + 1 ) { j 11 j 12 j j 21 j 22 j 23 j 31 j 32 j 33 } { j 11 j 12 j j 23 j 33 j } = ( 1 ) 2 j { j 21 j 22 j 23 j 12 j j 32 } { j 31 j 32 j 33 j j 11 j 21 } .
    30: 2.2 Transcendental Equations
    Conditions for the validity of the reversion process in are derived in Olver (1997b, pp. 14–16). …