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1: 9.13 Generalized Airy Functions
►Swanson and Headley (1967) define independent solutions A n ⁡ ( z ) and B n ⁡ ( z ) of (9.13.1) by …When n = 1 , A n ⁡ ( z ) and B n ⁡ ( z ) become Ai ⁡ ( z ) and Bi ⁡ ( z ) , respectively. ►Properties of A n ⁡ ( z ) and B n ⁡ ( z ) follow from the corresponding properties of the modified Bessel functions. … ►The distribution in ℂ and asymptotic properties of the zeros of A n ⁡ ( z ) , A n ⁡ ( z ) , B n ⁡ ( z ) , and B n ⁡ ( z ) are investigated in Swanson and Headley (1967) and Headley and Barwell (1975). … ►Their relations to the functions A n ⁡ ( z ) and B n ⁡ ( z ) are given by …
2: 25.19 Tables
►
  • Abramowitz and Stegun (1964) tabulates: ζ ⁡ ( n ) , n = 2 , 3 , 4 , , 20D (p. 811); Li 2 ⁡ ( 1 - x ) , x = 0 ⁢ ( .01 ) ⁢ 0.5 , 9D (p. 1005); f ⁡ ( θ ) , θ = 15 ∘ ⁢ ( 1 ∘ ) ⁢ 30 ∘ ⁢ ( 2 ∘ ) ⁢ 90 ∘ ⁢ ( 5 ∘ ) ⁢ 180 ∘ , f ⁡ ( θ ) + θ ⁢ ln ⁡ θ , θ = 0 ⁢ ( 1 ∘ ) ⁢ 15 ∘ , 6D (p. 1006). Here f ⁡ ( θ ) denotes Clausen’s integral, given by the right-hand side of (25.12.9).

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  • Morris (1979) tabulates Li 2 ⁡ ( x ) 25.12(i)) for ± x = 0.02 ⁢ ( .02 ) ⁢ 1 ⁢ ( .1 ) ⁢ 6 to 30D.

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  • Cloutman (1989) tabulates Γ ⁡ ( s + 1 ) ⁢ F s ⁡ ( x ) , where F s ⁡ ( x ) is the Fermi–Dirac integral (25.12.14), for s = - 1 2 , 1 2 , 3 2 , 5 2 , x = - 5 ⁢ ( .05 ) ⁢ 25 , to 12S.

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  • Fletcher et al. (1962, §22.1) lists many sources for earlier tables of ζ ⁡ ( s ) for both real and complex s . §22.133 gives sources for numerical values of coefficients in the Riemann–Siegel formula, §22.15 describes tables of values of ζ ⁡ ( s , a ) , and §22.17 lists tables for some Dirichlet L -functions for real characters. For tables of dilogarithms, polylogarithms, and Clausen’s integral see §§22.84–22.858.

  • 3: Bibliography O
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  • K. Okamoto (1987b) Studies on the Painlevé equations. II. Fifth Painlevé equation P V . Japan. J. Math. (N.S.) 13 (1), pp. 47–76.
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  • A. B. Olde Daalhuis (2005a) Hyperasymptotics for nonlinear ODEs. I. A Riccati equation. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2060), pp. 2503–2520.
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  • A. B. Olde Daalhuis (2005b) Hyperasymptotics for nonlinear ODEs. II. The first Painlevé equation and a second-order Riccati equation. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2062), pp. 3005–3021.
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  • F. W. J. Olver (1994a) Asymptotic expansions of the coefficients in asymptotic series solutions of linear differential equations. Methods Appl. Anal. 1 (1), pp. 1–13.
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  • C. Osácar, J. Palacián, and M. Palacios (1995) Numerical evaluation of the dilogarithm of complex argument. Celestial Mech. Dynam. Astronom. 62 (1), pp. 93–98.
  • 4: Mark J. Ablowitz
    ►Their similarity solutions lead to special ODEs which have the Painlevé property; i. …ODEs which do not have moveable branch point singularities. ODEs with the Painlevé property contain the well-known Painlevé equations which are special second order scalar equations; their solutions are often called Painlevé transcendents. …
    5: 9.10 Integrals
    ►
    9.10.8 z ⁢ w ⁡ ( z ) ⁢ d z = w ⁡ ( z ) ,
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    9.10.9 z 2 ⁢ w ⁡ ( z ) ⁢ d z = z ⁢ w ⁡ ( z ) - w ⁡ ( z ) ,
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    9.10.10 z n + 3 ⁢ w ⁡ ( z ) ⁢ d z = z n + 2 ⁢ w ⁡ ( z ) - ( n + 2 ) ⁢ z n + 1 ⁢ w ⁡ ( z ) + ( n + 1 ) ⁢ ( n + 2 ) ⁢ z n ⁢ w ⁡ ( z ) ⁢ d z , n = 0 , 1 , 2 , .
    ►For the confluent hypergeometric function F 1 1 and the incomplete gamma function Γ see §§13.1, 13.2, and 8.2(i). … ►For further integrals, including the Airy transform, see §9.11(iv), Widder (1979), Prudnikov et al. (1990, §1.8.1), Prudnikov et al. (1992a, pp. 405–413), Prudnikov et al. (1992b, §4.3.25), Vallée and Soares (2010, Chapters 3, 4).
    6: Bibliography L
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  • W. Lay and S. Yu. Slavyanov (1998) The central two-point connection problem for the Heun class of ODEs. J. Phys. A 31 (18), pp. 4249–4261.
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  • J. L. López, P. Pagola, and E. Pérez Sinusía (2013a) Asymptotics of the first Appell function F 1 with large parameters II. Integral Transforms Spec. Funct. 24 (12), pp. 982–999.
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  • J. L. López, P. Pagola, and E. Pérez Sinusía (2013b) Asymptotics of the first Appell function F 1 with large parameters. Integral Transforms Spec. Funct. 24 (9), pp. 715–733.
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  • J. L. López and E. Pérez Sinusía (2014) New series expansions for the confluent hypergeometric function M ⁢ ( a , b , z ) . Appl. Math. Comput. 235, pp. 26–31.
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  • 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.
  • 7: 22.19 Physical Applications
    ►This formulation gives the bounded and unbounded solutions from the same formula (22.19.3), for k 1 and k 1 , respectively. … ►
    Case I: V ⁡ ( x ) = 1 2 ⁢ x 2 + 1 4 ⁢ β ⁢ x 4
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    Case II: V ⁡ ( x ) = 1 2 ⁢ x 2 - 1 4 ⁢ β ⁢ x 4
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    Case III: V ⁡ ( x ) = - 1 2 ⁢ x 2 + 1 4 ⁢ β ⁢ x 4
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    §22.19(iii) Nonlinear ODEs and PDEs
    8: 28.9 Zeros
    ►For real q each of the functions ce 2 ⁢ n ⁡ ( z , q ) , se 2 ⁢ n + 1 ⁡ ( z , q ) , ce 2 ⁢ n + 1 ⁡ ( z , q ) , and se 2 ⁢ n + 2 ⁡ ( z , q ) has exactly n zeros in 0 < z < 1 2 ⁢ π . …For q the zeros of ce 2 ⁢ n ⁡ ( z , q ) and se 2 ⁢ n + 1 ⁡ ( z , q ) approach asymptotically the zeros of He 2 ⁢ n ⁡ ( q 1 / 4 ⁢ ( π - 2 ⁢ z ) ) , and the zeros of ce 2 ⁢ n + 1 ⁡ ( z , q ) and se 2 ⁢ n + 2 ⁡ ( z , q ) approach asymptotically the zeros of He 2 ⁢ n + 1 ⁡ ( q 1 / 4 ⁢ ( π - 2 ⁢ z ) ) . …There are no zeros within the strip | ⁡ z | < 1 2 ⁢ π other than those on the real and imaginary axes. ►For further details see McLachlan (1947, pp. 234–239) and Meixner and Schäfke (1954, §§2.331, 2.8, 2.81, and 2.85).
    9: Bibliography G
    ►
  • B. Gambier (1910) Sur les équations différentielles du second ordre et du premier degré dont l’intégrale générale est a points critiques fixes. Acta Math. 33 (1), pp. 1–55.
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  • A. Gil, J. Segura, and N. M. Temme (2006a) Computing the real parabolic cylinder functions U ⁢ ( a , x ) , V ⁢ ( a , x ) . ACM Trans. Math. Software 32 (1), pp. 70–101.
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  • A. Gil and J. Segura (2003) Computing the zeros and turning points of solutions of second order homogeneous linear ODEs. SIAM J. Numer. Anal. 41 (3), pp. 827–855.
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  • R. L. Graham, M. Grötschel, and L. Lovász (Eds.) (1995) Handbook of Combinatorics. Vols. 1, 2. Elsevier Science B.V., Amsterdam.
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  • M. B. Green, J. H. Schwarz, and E. Witten (1988a) Superstring Theory: Introduction, Vol. 1. 2nd edition, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge.
  • 10: 9.2 Differential Equation
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    9.2.1 d 2 w d z 2 = z ⁢ w .
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    9.2.2 w = Ai ⁡ ( z ) , Bi ⁡ ( z ) , Ai ⁡ ( z ⁢ e ∓ 2 ⁢ π ⁢ i / 3 ) .
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    9.2.4 Ai ⁡ ( 0 ) = - 1 3 1 / 3 ⁢ Γ ⁡ ( 1 3 ) = - 0.25881 94037 ⁢ ,
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    9.2.5 Bi ⁡ ( 0 ) = 1 3 1 / 6 ⁢ Γ ⁡ ( 2 3 ) = 0.61492 66274 ⁢ ,
    ► W = ( 1 / w ) ⁢ d w / d z , where w is any nontrivial solution of (9.2.1). …