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11: 11 Struve and Related Functions
Chapter 11 Struve and Related Functions
12: Bibliography L
  • T. M. Larsen, D. Erricolo, and P. L. E. Uslenghi (2009) New method to obtain small parameter power series expansions of Mathieu radial and angular functions. Math. Comp. 78 (265), pp. 255–274.
  • 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.
  • 13: 3.9 Acceleration of Convergence
    For further information on the epsilon algorithm see Brezinski and Redivo Zaglia (1991, pp. 78–95). …
    Table 3.9.1: Shanks’ transformation for s n = j = 1 n ( 1 ) j + 1 j 2 .
    n t n , 2 t n , 4 t n , 6 t n , 8 t n , 10
    2 0.82111 11111 11 0.82243 44785 14 0.82246 61821 45 0.82246 70102 48 0.82246 70327 79
    14: 33.17 Recurrence Relations and Derivatives
    15: Bibliography P
  • J. B. Parkinson (1969) Optical properties of layer antiferromagnets with K 2 NiF 4 structure. J. Phys. C: Solid State Physics 2 (11), pp. 2012–2021.
  • R. Piessens and M. Branders (1984) Algorithm 28. Algorithm for the computation of Bessel function integrals. J. Comput. Appl. Math. 11 (1), pp. 119–137.
  • G. Pólya (1949) Remarks on computing the probability integral in one and two dimensions. In Proceedings of the Berkeley Symposium on Mathematical Statistics and Probability, 1945, 1946, pp. 63–78.
  • 16: Bibliography C
  • L. Carlitz (1960) Note on Nörlund’s polynomial B n ( z ) . Proc. Amer. Math. Soc. 11 (3), pp. 452–455.
  • C. Chiccoli, S. Lorenzutta, and G. Maino (1988) On the evaluation of generalized exponential integrals E v ( x ) . J. Comput. Phys. 78 (2), pp. 278–287.
  • 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.
  • F. Cooper, A. Khare, and A. Saxena (2006) Exact elliptic compactons in generalized Korteweg-de Vries equations. Complexity 11 (6), pp. 30–34.
  • 17: 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).
  • D. R. Herrick and S. O’Connor (1998) Inverse virial symmetry of diatomic potential curves. J. Chem. Phys. 109 (1), pp. 11–19.
  • H. W. Hethcote (1970) Error bounds for asymptotic approximations of zeros of Hankel functions occurring in diffraction problems. J. Mathematical Phys. 11 (8), pp. 2501–2504.
  • G. W. Hill (1970) Algorithm 395: Student’s t-distribution. Comm. ACM 13 (10), pp. 617–619.
  • 18: 25.9 Asymptotic Approximations
    25.9.2 χ ( s ) π s 1 2 Γ ( 1 2 1 2 s ) / Γ ( 1 2 s ) .
    19: 34.6 Definition: 9 j Symbol
    34.6.1 { j 11 j 12 j 13 j 21 j 22 j 23 j 31 j 32 j 33 } = all  m r s ( 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 ) ( j 13 j 23 j 33 m 13 m 23 m 33 ) ,
    34.6.2 { j 11 j 12 j 13 j 21 j 22 j 23 j 31 j 32 j 33 } = j ( 1 ) 2 j ( 2 j + 1 ) { j 11 j 21 j 31 j 32 j 33 j } { j 12 j 22 j 32 j 21 j j 23 } { j 13 j 23 j 33 j j 11 j 12 } .
    20: 13.30 Tables
  • Slater (1960) tabulates M ( a , b , x ) for a = 1 ( .1 ) 1 , b = 0.1 ( .1 ) 1 , and x = 0.1 ( .1 ) 10 , 7–9S; M ( a , b , 1 ) for a = 11 ( .2 ) 2 and b = 4 ( .2 ) 1 , 7D; the smallest positive x -zero of M ( a , b , x ) for a = 4 ( .1 ) 0.1 and b = 0.1 ( .1 ) 2.5 , 7D.

  • Zhang and Jin (1996, pp. 411–423) tabulates M ( a , b , x ) and U ( a , b , x ) for a = 5 ( .5 ) 5 , b = 0.5 ( .5 ) 5 , and x = 0.1 , 1 , 5 , 10 , 20 , 30 , 8S (for M ( a , b , x ) ) and 7S (for U ( a , b , x ) ).