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11: Bibliography E
  • C. Eckart (1930) The penetration of a potential barrier by electrons. Phys. Rev. 35 (11), pp. 1303–1309.
  • A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi (1953b) Higher Transcendental Functions. Vol. II. McGraw-Hill Book Company, Inc., New York-Toronto-London.
  • F. H. L. Essler, H. Frahm, A. R. Its, and V. E. Korepin (1996) Painlevé transcendent describes quantum correlation function of the X X Z antiferromagnet away from the free-fermion point. J. Phys. A 29 (17), pp. 5619–5626.
  • L. Euler (1768) Institutiones Calculi Integralis. Opera Omnia (1), Vol. 11, pp. 110–113.
  • W. N. Everitt (2008) Note on the X 1 -Laguerre orthogonal polynomials.
  • 12: 4.17 Special Values and Limits
    Table 4.17.1: Trigonometric functions: values at multiples of 1 12 π .
    θ sin θ cos θ tan θ csc θ sec θ cot θ
    11 π / 12 1 4 2 ( 3 1 ) 1 4 2 ( 3 + 1 ) ( 2 3 ) 2 ( 3 + 1 ) 2 ( 3 1 ) ( 2 + 3 )
    4.17.1 lim z 0 sin z z = 1 ,
    4.17.2 lim z 0 tan z z = 1 .
    4.17.3 lim z 0 1 cos z z 2 = 1 2 .
    13: Bibliography K
  • E. L. Kaplan (1948) Auxiliary table for the incomplete elliptic integrals. J. Math. Physics 27, pp. 1136.
  • M. K. Kerimov and S. L. Skorokhodov (1985b) Calculation of the complex zeros of Hankel functions and their derivatives. Zh. Vychisl. Mat. i Mat. Fiz. 25 (11), pp. 1628–1643, 1741.
  • S. K. Kim (1972) The asymptotic expansion of a hypergeometric function F 2 2 ( 1 , α ; ρ 1 , ρ 2 ; z ) . Math. Comp. 26 (120), pp. 963.
  • 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.
  • K. S. Kölbig (1968) Algorithm 327: Dilogarithm [S22]. Comm. ACM 11 (4), pp. 270–271.
  • 14: Staff
  • Michael V. Berry, University of Bristol, Bristol, Chap. 36

  • Richard B. Paris, University of Abertay, Chaps. 8, 11

  • Hans Volkmer, University of Wisconsin, Milwaukee, Chaps. 29, 30

  • Richard B. Paris, University of Abertay Dundee, for Chaps. 8, 11 (deceased)

  • Hans Volkmer, University of Wisconsin–Milwaukee, for Chaps. 29, 30

  • 15: Bibliography H
  • R. S. Heller (1976) 25D Table of the First One Hundred Values of j 0 , s , J 1 ( j 0 , s ) , j 1 , s , J 0 ( j 1 , s ) = J 0 ( j 0 , s + 1 ) , j 1 , s , J 1 ( j 1 , s ) . Technical report Department of Physics, Worcester Polytechnic Institute, Worcester, MA.
  • 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.
  • K. Horata (1989) An explicit formula for Bernoulli numbers. Rep. Fac. Sci. Technol. Meijo Univ. 29, pp. 1–6.
  • 16: Bibliography L
  • A. Leitner and J. Meixner (1960) Eine Verallgemeinerung der Sphäroidfunktionen. Arch. Math. 11, pp. 29–39.
  • M. Lerch (1887) Note sur la fonction 𝔎 ( w , x , s ) = k = 0 e 2 k π i x ( w + k ) s . Acta Math. 11 (1-4), pp. 19–24 (French).
  • 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.
  • 17: 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.
  • D. Dominici, S. J. Johnston, and K. Jordaan (2013) Real zeros of F 1 2 hypergeometric polynomials. J. Comput. Appl. Math. 247, pp. 152–161.
  • 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. 1129.
  • 18: 26.16 Multiset Permutations
    Let S = { 1 a 1 , 2 a 2 , , n a n } be the multiset that has a j copies of j , 1 j n . 𝔖 S denotes the set of permutations of S for all distinct orderings of the a 1 + a 2 + + a n integers. The number of elements in 𝔖 S is the multinomial coefficient (§26.4) ( a 1 + a 2 + + a n a 1 , a 2 , , a n ) . … The q -multinomial coefficient is defined in terms of Gaussian polynomials (§26.9(ii)) by …and again with S = { 1 a 1 , 2 a 2 , , n a n } we have …
    19: 34.1 Special Notation
    ( j 1 j 2 j 3 m 1 m 2 m 3 ) ,
    { j 1 j 2 j 3 l 1 l 2 l 3 } ,
    { j 11 j 12 j 13 j 21 j 22 j 23 j 31 j 32 j 33 } .
    An often used alternative to the 3 j symbol is the Clebsch–Gordan coefficient
    34.1.1 ( j 1 m 1 j 2 m 2 | j 1 j 2 j 3 m 3 ) = ( 1 ) j 1 j 2 + m 3 ( 2 j 3 + 1 ) 1 2 ( j 1 j 2 j 3 m 1 m 2 m 3 ) ;
    20: Bibliography G
  • W. Gautschi (1966) Algorithm 292: Regular Coulomb wave functions. Comm. ACM 9 (11), pp. 793–795.
  • W. Gautschi (1969) Algorithm 363: Complex error function. Comm. ACM 12 (11), pp. 635.
  • A. Gervois and H. Navelet (1984) Some integrals involving three Bessel functions when their arguments satisfy the triangle inequalities. J. Math. Phys. 25 (11), pp. 3350–3356.
  • H. W. Gould (1960) Stirling number representation problems. Proc. Amer. Math. Soc. 11 (3), pp. 447–451.
  • V. I. Gromak (1975) Theory of Painlevé’s equations. Differ. Uravn. 11 (11), pp. 373–376 (Russian).