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31: 7.23 Tables
  • Abramowitz and Stegun (1964, Chapter 7) includes erf x , ( 2 / π ) e x 2 , x [ 0 , 2 ] , 10D; ( 2 / π ) e x 2 , x [ 2 , 10 ] , 8S; x e x 2 erfc x , x 2 [ 0 , 0.25 ] , 7D; 2 n Γ ( 1 2 n + 1 ) i n erfc ( x ) , n = 1 ( 1 ) 6 , 10 , 11 , x [ 0 , 5 ] , 6S; F ( x ) , x [ 0 , 2 ] , 10D; x F ( x ) , x 2 [ 0 , 0.25 ] , 9D; C ( x ) , S ( x ) , x [ 0 , 5 ] , 7D; f ( x ) , g ( x ) , x [ 0 , 1 ] , x 1 [ 0 , 1 ] , 15D.

  • Zhang and Jin (1996, pp. 637, 639) includes ( 2 / π ) e x 2 , erf x , x = 0 ( .02 ) 1 ( .04 ) 3 , 8D; C ( x ) , S ( x ) , x = 0 ( .2 ) 10 ( 2 ) 100 ( 100 ) 500 , 8D.

  • Fettis et al. (1973) gives the first 100 zeros of erf z and w ( z ) (the table on page 406 of this reference is for w ( z ) , not for erfc z ), 11S.

  • Zhang and Jin (1996, p. 642) includes the first 10 zeros of erf z , 9D; the first 25 distinct zeros of C ( z ) and S ( z ) , 8S.

  • 32: Bibliography W
  • R. J. Wells (1999) Rapid approximation to the Voigt/Faddeeva function and its derivatives. J. Quant. Spect. and Rad. Transfer 62 (1), pp. 29–48.
  • J. A. Wilson (1980) Some hypergeometric orthogonal polynomials. SIAM J. Math. Anal. 11 (4), pp. 690–701.
  • G. Wolf (2008) On the asymptotic behavior of the Fourier coefficients of Mathieu functions. J. Res. Nat. Inst. Standards Tech. 113 (1), pp. 11–15.
  • R. Wong and M. Wyman (1974) The method of Darboux. J. Approximation Theory 10 (2), pp. 159–171.
  • E. M. Wright (1940b) The generalized Bessel function of order greater than one. Quart. J. Math., Oxford Ser. 11, pp. 36–48.
  • 33: Bibliography F
  • V. N. Faddeyeva and N. M. Terent’ev (1961) Tables of Values of the Function w ( z ) = e z 2 ( 1 + 2 i π 1 / 2 0 z e t 2 𝑑 t ) for Complex Argument. Edited by V. A. Fok; translated from the Russian by D. G. Fry. Mathematical Tables Series, Vol. 11, Pergamon Press, Oxford.
  • N. Fleury and A. Turbiner (1994) Polynomial relations in the Heisenberg algebra. J. Math. Phys. 35 (11), pp. 6144–6149.
  • A. S. Fokas and M. J. Ablowitz (1982) On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23 (11), pp. 2033–2042.
  • P. J. Forrester and N. S. Witte (2004) Application of the τ -function theory of Painlevé equations to random matrices: P VI , the JUE, CyUE, cJUE and scaled limits. Nagoya Math. J. 174, pp. 29–114.
  • L. W. Fullerton (1972) Algorithm 435: Modified incomplete gamma function. Comm. ACM 15 (11), pp. 993–995.
  • 34: 8.26 Tables
  • Pearson (1968) tabulates I x ( a , b ) for x = 0.01 ( .01 ) 1 , a , b = 0.5 ( .5 ) 11 ( 1 ) 50 , with b a , to 7D.

  • Zhang and Jin (1996, Table 3.9) tabulates I x ( a , b ) for x = 0 ( .05 ) 1 , a = 0.5 , 1 , 3 , 5 , 10 , b = 1 , 10 to 8D.

  • Pagurova (1961) tabulates E n ( x ) for n = 0 ( 1 ) 20 , x = 0 ( .01 ) 2 ( .1 ) 10 to 4-9S; e x E n ( x ) for n = 2 ( 1 ) 10 , x = 10 ( .1 ) 20 to 7D; e x E p ( x ) for p = 0 ( .1 ) 1 , x = 0.01 ( .01 ) 7 ( .05 ) 12 ( .1 ) 20 to 7S or 7D.

  • Stankiewicz (1968) tabulates E n ( x ) for n = 1 ( 1 ) 10 , x = 0.01 ( .01 ) 5 to 7D.

  • Zhang and Jin (1996, Table 19.1) tabulates E n ( x ) for n = 1 , 2 , 3 , 5 , 10 , 15 , 20 , x = 0 ( .1 ) 1 , 1.5 , 2 , 3 , 5 , 10 , 20 , 30 , 50 , 100 to 7D or 8S.

  • 35: 5.22 Tables
    Abramowitz and Stegun (1964, Chapter 6) tabulates Γ ( x ) , ln Γ ( x ) , ψ ( x ) , and ψ ( x ) for x = 1 ( .005 ) 2 to 10D; ψ ′′ ( x ) and ψ ( 3 ) ( x ) for x = 1 ( .01 ) 2 to 10D; Γ ( n ) , 1 / Γ ( n ) , Γ ( n + 1 2 ) , ψ ( n ) , log 10 Γ ( n ) , log 10 Γ ( n + 1 3 ) , log 10 Γ ( n + 1 2 ) , and log 10 Γ ( n + 2 3 ) for n = 1 ( 1 ) 101 to 8–11S; Γ ( n + 1 ) for n = 100 ( 100 ) 1000 to 20S. Zhang and Jin (1996, pp. 67–69 and 72) tabulates Γ ( x ) , 1 / Γ ( x ) , Γ ( x ) , ln Γ ( x ) , ψ ( x ) , ψ ( x ) , ψ ( x ) , and ψ ( x ) for x = 0 ( .1 ) 5 to 8D or 8S; Γ ( n + 1 ) for n = 0 ( 1 ) 100 ( 10 ) 250 ( 50 ) 500 ( 100 ) 3000 to 51S. … Abramowitz and Stegun (1964, Chapter 6) tabulates ln Γ ( x + i y ) for x = 1 ( .1 ) 2 , y = 0 ( .1 ) 10 to 12D. …Zhang and Jin (1996, pp. 70, 71, and 73) tabulates the real and imaginary parts of Γ ( x + i y ) , ln Γ ( x + i y ) , and ψ ( x + i y ) for x = 0.5 , 1 , 5 , 10 , y = 0 ( .5 ) 10 to 8S.
    36: 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.
  • 37: 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 } .
    38: 6.19 Tables
  • Abramowitz and Stegun (1964, Chapter 5) includes x 1 Si ( x ) , x 2 Cin ( x ) , x 1 Ein ( x ) , x 1 Ein ( x ) , x = 0 ( .01 ) 0.5 ; Si ( x ) , Ci ( x ) , Ei ( x ) , E 1 ( x ) , x = 0.5 ( .01 ) 2 ; Si ( x ) , Ci ( x ) , x e x Ei ( x ) , x e x E 1 ( x ) , x = 2 ( .1 ) 10 ; x f ( x ) , x 2 g ( x ) , x e x Ei ( x ) , x e x E 1 ( x ) , x 1 = 0 ( .005 ) 0.1 ; Si ( π x ) , Cin ( π x ) , x = 0 ( .1 ) 10 . Accuracy varies but is within the range 8S–11S.

  • Zhang and Jin (1996, pp. 690–692) includes the real and imaginary parts of E 1 ( z ) , ± x = 0.5 , 1 , 3 , 5 , 10 , 15 , 20 , 50 , 100 , y = 0 ( .5 ) 1 ( 1 ) 5 ( 5 ) 30 , 50 , 100 , 8S.

  • 39: 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.
  • Cunningham Project (website)
  • 40: Bibliography M
  • M. Mazzocco (2001a) Rational solutions of the Painlevé VI equation. J. Phys. A 34 (11), pp. 2281–2294.
  • T. Morita (1978) Calculation of the complete elliptic integrals with complex modulus. Numer. Math. 29 (2), pp. 233–236.
  • L. Moser and M. Wyman (1958b) Stirling numbers of the second kind. Duke Math. J. 25 (1), pp. 29–43.
  • D. Müller, B. G. Kelly, and J. J. O’Brien (1994) Spheroidal eigenfunctions of the tidal equation. Phys. Rev. Lett. 73 (11), pp. 1557–1560.
  • L. A. Muraveĭ (1976) Zeros of the function A i ( z ) σ A i ( z ) . Differential Equations 11, pp. 797–811.