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1: 4.40 Integrals
4.40.1 sinh x d x = cosh x ,
4.40.2 cosh x d x = sinh x ,
4.40.5 sech x d x = gd ( x ) .
Extensive compendia of indefinite and definite integrals of hyperbolic functions include Apelblat (1983, pp. 96–109), Bierens de Haan (1939), Gröbner and Hofreiter (1949, pp. 139–160), Gröbner and Hofreiter (1950, pp. 160–167), Gradshteyn and Ryzhik (2000, Chapters 2–4), and Prudnikov et al. (1986a, §§1.4, 1.8, 2.4, 2.8).
2: 4.26 Integrals
4.26.1 sin x d x = cos x ,
4.26.2 cos x d x = sin x .
4.26.11 0 π sin 2 ( n t ) d t = 0 π cos 2 ( n t ) d t = 1 2 π , n 0 .
4.26.13 0 sin ( t 2 ) d t = 0 cos ( t 2 ) d t = 1 2 π 2 .
Extensive compendia of indefinite and definite integrals of trigonometric and inverse trigonometric functions include Apelblat (1983, pp. 48–109), Bierens de Haan (1939), Gradshteyn and Ryzhik (2000, Chapters 2–4), Gröbner and Hofreiter (1949, pp. 116–139), Gröbner and Hofreiter (1950, pp. 94–160), and Prudnikov et al. (1986a, §§1.5, 1.7, 2.5, 2.7).
3: Bibliography L
  • J. Lagrange (1770) Démonstration d’un Théoréme d’Arithmétique. Nouveau Mém. Acad. Roy. Sci. Berlin, pp. 123–133 (French).
  • D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.
  • D. W. Lozier and F. W. J. Olver (1993) Airy and Bessel Functions by Parallel Integration of ODEs. In Proceedings of the Sixth SIAM Conference on Parallel Processing for Scientific Computing, R. F. Sincovec, D. E. Keyes, M. R. Leuze, L. R. Petzold, and D. A. Reed (Eds.), Philadelphia, PA, pp. 530–538.
  • Y. L. Luke (1970) Further approximations for elliptic integrals. Math. Comp. 24 (109), pp. 191–198.
  • J. N. Lyness (1985) Integrating some infinite oscillating tails. J. Comput. Appl. Math. 12/13, pp. 109–117.
  • 4: 27.2 Functions
    27.2.9 d ( n ) = d | n 1
    It is the special case k = 2 of the function d k ( n ) that counts the number of ways of expressing n as the product of k factors, with the order of factors taken into account.
    27.2.10 σ α ( n ) = d | n d α ,
    Note that σ 0 ( n ) = d ( n ) . … Table 27.2.2 tabulates the Euler totient function ϕ ( n ) , the divisor function d ( n ) ( = σ 0 ( n ) ), and the sum of the divisors σ ( n ) ( = σ 1 ( n ) ), for n = 1 ( 1 ) 52 . …
    5: Bibliography E
  • M. Edwards, D. A. Griggs, P. L. Holman, C. W. Clark, S. L. Rolston, and W. D. Phillips (1999) Properties of a Raman atom-laser output coupler. J. Phys. B 32 (12), pp. 2935–2950.
  • D. Elliott (1971) Uniform asymptotic expansions of the Jacobi polynomials and an associated function. Math. Comp. 25 (114), pp. 309–315.
  • D. Elliott (1998) The Euler-Maclaurin formula revisited. J. Austral. Math. Soc. Ser. B 40 (E), pp. E27–E76 (electronic).
  • A. Erdélyi (1956) Asymptotic Expansions. Dover Publications Inc., New York.
  • W. D. Evans, W. N. Everitt, K. H. Kwon, and L. L. Littlejohn (1993) Real orthogonalizing weights for Bessel polynomials. J. Comput. Appl. Math. 49 (1-3), pp. 51–57.
  • 6: Bibliography Z
  • D. Zagier (1989) The Dilogarithm Function in Geometry and Number Theory. In Number Theory and Related Topics (Bombay, 1988), R. Askey and others (Eds.), Tata Inst. Fund. Res. Stud. Math., Vol. 12, pp. 231–249.
  • D. Zagier (1998) A modified Bernoulli number. Nieuw Arch. Wisk. (4) 16 (1-2), pp. 63–72.
  • A. Zarzo, J. S. Dehesa, and R. J. Yañez (1995) Distribution of zeros of Gauss and Kummer hypergeometric functions. A semiclassical approach. Ann. Numer. Math. 2 (1-4), pp. 457–472.
  • D. Zeilberger and D. M. Bressoud (1985) A proof of Andrews’ q -Dyson conjecture. Discrete Math. 54 (2), pp. 201–224.
  • D. G. Zill and B. C. Carlson (1970) Symmetric elliptic integrals of the third kind. Math. Comp. 24 (109), pp. 199–214.
  • 7: Bibliography
  • P. Appell and J. Kampé de Fériet (1926) Fonctions hypergéométriques et hypersphériques. Polynomes d’Hermite. Gauthier-Villars, Paris.
  • V. I. Arnol’d, S. M. Guseĭn-Zade, and A. N. Varchenko (1988) Singularities of Differentiable Maps. Vol. II. Birkhäuser, Boston-Berlin.
  • V. I. Arnol’d (1975) Critical points of smooth functions, and their normal forms. Uspehi Mat. Nauk 30 (5(185)), pp. 3–65 (Russian).
  • V. I. Arnol’d (1986) Catastrophe Theory. 2nd edition, Springer-Verlag, Berlin.
  • V. I. Arnol’d (1992) Catastrophe Theory. 3rd edition, Springer-Verlag, Berlin.
  • 8: 4.20 Derivatives and Differential Equations
    4.20.1 d d z sin z = cos z ,
    4.20.2 d d z cos z = sin z ,
    4.20.3 d d z tan z = sec 2 z ,
    4.20.6 d d z cot z = csc 2 z ,
    4.20.9 d 2 w d z 2 + a 2 w = 0 ,
    9: 4.34 Derivatives and Differential Equations
    4.34.1 d d z sinh z = cosh z ,
    4.34.7 d 2 w d z 2 a 2 w = 0 ,
    4.34.8 ( d w d z ) 2 a 2 w 2 = 1 ,
    4.34.9 ( d w d z ) 2 a 2 w 2 = 1 ,
    4.34.10 d w d z + a 2 w 2 = 1 ,
    10: 22.13 Derivatives and Differential Equations
    22.13.1 ( d d z sn ( z , k ) ) 2 = ( 1 sn 2 ( z , k ) ) ( 1 k 2 sn 2 ( z , k ) ) ,
    22.13.2 ( d d z cn ( z , k ) ) 2 = ( 1 cn 2 ( z , k ) ) ( k 2 + k 2 cn 2 ( z , k ) ) ,
    22.13.3 ( d d z dn ( z , k ) ) 2 = ( 1 dn 2 ( z , k ) ) ( dn 2 ( z , k ) k 2 ) .
    22.13.7 ( d d z dc ( z , k ) ) 2 = ( dc 2 ( z , k ) 1 ) ( dc 2 ( z , k ) k 2 ) ,
    22.13.10 ( d d z ns ( z , k ) ) 2 = ( ns 2 ( z , k ) k 2 ) ( ns 2 ( z , k ) 1 ) ,