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double asymptotic properties

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11: Bibliography F
  • B. R. Fabijonas (2004) Algorithm 838: Airy functions. ACM Trans. Math. Software 30 (4), pp. 491–501.
  • FDLIBM (free C library)
  • C. Ferreira and J. L. López (2001) An asymptotic expansion of the double gamma function. J. Approx. Theory 111 (2), pp. 298–314.
  • A. S. Fokas and Y. C. Yortsos (1981) The transformation properties of the sixth Painlevé equation and one-parameter families of solutions. Lett. Nuovo Cimento (2) 30 (17), pp. 539–544.
  • R. C. Forrey (1997) Computing the hypergeometric function. J. Comput. Phys. 137 (1), pp. 79–100.
  • 12: Bibliography B
  • K. L. Bell and N. S. Scott (1980) Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.
  • T. Bountis, H. Segur, and F. Vivaldi (1982) Integrable Hamiltonian systems and the Painlevé property. Phys. Rev. A (3) 25 (3), pp. 1257–1264.
  • W. Bühring (1994) The double confluent Heun equation: Characteristic exponent and connection formulae. Methods Appl. Anal. 1 (3), pp. 348–370.
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  • J. L. Burchnall and T. W. Chaundy (1941) Expansions of Appell’s double hypergeometric functions. II. Quart. J. Math., Oxford Ser. 12, pp. 112–128.
  • 13: Bibliography O
  • A. M. Odlyzko (1995) Asymptotic Enumeration Methods. In Handbook of Combinatorics, Vol. 2, L. Lovász, R. L. Graham, and M. Grötschel (Eds.), pp. 1063–1229.
  • A. B. Olde Daalhuis (1998c) On the resurgence properties of the uniform asymptotic expansion of the incomplete gamma function. Methods Appl. Anal. 5 (4), pp. 425–438.
  • A. B. Olde Daalhuis (2000) On the asymptotics for late coefficients in uniform asymptotic expansions of integrals with coalescing saddles. Methods Appl. Anal. 7 (4), pp. 727–745.
  • F. W. J. Olver and J. M. Smith (1983) Associated Legendre functions on the cut. J. Comput. Phys. 51 (3), pp. 502–518.
  • 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.
  • 14: Bibliography R
  • S. Ramanujan (1921) Congruence properties of partitions. Math. Z. 9 (1-2), pp. 147–153.
  • S. Ramanujan (1927) Some properties of Bernoulli’s numbers (J. Indian Math. Soc. 3 (1911), 219–234.). In Collected Papers,
  • J. Raynal (1979) On the definition and properties of generalized 6 - j  symbols. J. Math. Phys. 20 (12), pp. 2398–2415.
  • W. P. Reinhardt (2018) Universality properties of Gaussian quadrature, the derivative rule, and a novel approach to Stieltjes inversion.
  • M. Robnik (1980) An extremum property of the n -dimensional sphere. J. Phys. A 13 (10), pp. L349–L351.
  • 15: Bibliography G
  • E. A. Galapon and K. M. L. Martinez (2014) Exactification of the Poincaré asymptotic expansion of the Hankel integral: spectacularly accurate asymptotic expansions and non-asymptotic scales. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470 (2162), pp. 20130529, 16.
  • I. M. Gel’fand and G. E. Shilov (1964) Generalized Functions. Vol. 1: Properties and Operations. Academic Press, New York.
  • Z. Gong, L. Zejda, W. Dappen, and J. M. Aparicio (2001) Generalized Fermi-Dirac functions and derivatives: Properties and evaluation. Comput. Phys. Comm. 136 (3), pp. 294–309.
  • B. Grammaticos, A. Ramani, and V. Papageorgiou (1991) Do integrable mappings have the Painlevé property?. Phys. Rev. Lett. 67 (14), pp. 1825–1828.
  • P. Groeneboom and D. R. Truax (2000) A monotonicity property of the power function of multivariate tests. Indag. Math. (N.S.) 11 (2), pp. 209–218.
  • 16: 10.18 Modulus and Phase Functions
    §10.18(ii) Basic Properties
    §10.18(iii) Asymptotic Expansions for Large Argument
    10.18.19 N ν 2 ( x ) 2 π x ( 1 1 2 μ 3 ( 2 x ) 2 1 2 4 ( μ 1 ) ( μ 45 ) ( 2 x ) 4 ) ,
    10.18.20 ( 2 k 3 ) !! ( 2 k ) !! ( μ 1 ) ( μ 9 ) ( μ ( 2 k 3 ) 2 ) ( μ ( 2 k + 1 ) ( 2 k 1 ) 2 ) ( 2 x ) 2 k , k 2 ,
    In (10.18.17) and (10.18.18) the remainder after n terms does not exceed the ( n + 1 ) th term in absolute value and is of the same sign, provided that n > ν 1 2 for (10.18.17) and 3 2 ν 3 2 for (10.18.18).
    17: 28.31 Equations of Whittaker–Hill and Ince
    ℎ𝑠 2 n + 2 2 m + 2 ( z , ξ ) = ( 1 ) m ℎ𝑠 2 n + 2 2 m + 2 ( 1 2 π z , ξ ) .
    More important are the double orthogonality relations for p 1 p 2 or m 1 m 2 or both, given by …
    Asymptotic Behavior
    For ξ > 0 , the functions ℎ𝑐 p m ( z , ξ ) , ℎ𝑠 p m ( z , ξ ) behave asymptotically as multiples of exp ( 1 4 ξ cos ( 2 z ) ) ( cos z ) p as z ± i . … For ξ > 0 , the functions ℎ𝑐 p m ( z , ξ ) , ℎ𝑠 p m ( z , ξ ) behave asymptotically as multiples of exp ( 1 4 ξ cos ( 2 z ) ) ( cos z ) p 2 as z 1 2 π ± i . …
    18: Bibliography H
  • D. R. Hartree (1936) Some properties and applications of the repeated integrals of the error function. Proc. Manchester Lit. Philos. Soc. 80, pp. 85–102.
  • I. D. Hill (1973) Algorithm AS66: The normal integral. Appl. Statist. 22 (3), pp. 424–427.
  • M. Hiyama and H. Nakamura (1997) Two-center Coulomb functions. Comput. Phys. Comm. 103 (2-3), pp. 209–216.
  • C. J. Howls and A. B. Olde Daalhuis (1999) On the resurgence properties of the uniform asymptotic expansion of Bessel functions of large order. Proc. Roy. Soc. London Ser. A 455, pp. 3917–3930.
  • J. Humblet (1984) Analytical structure and properties of Coulomb wave functions for real and complex energies. Ann. Physics 155 (2), pp. 461–493.
  • 19: Bibliography C
  • J. B. Campbell (1984) Determination of ν -zeros of Hankel functions. Comput. Phys. Comm. 32 (3), pp. 333–339.
  • B. C. Carlson (2006a) Some reformulated properties of Jacobian elliptic functions. J. Math. Anal. Appl. 323 (1), pp. 522–529.
  • J. M. Carnicer, E. Mainar, and J. M. Peña (2020) Stability properties of disk polynomials. Numer. Algorithms.
  • J. A. Christley and I. J. Thompson (1994) CRCWFN: Coupled real Coulomb wavefunctions. Comput. Phys. Comm. 79 (1), pp. 143–155.
  • P. A. Clarkson and K. Jordaan (2018) Properties of generalized Freud polynomials. J. Approx. Theory 225, pp. 148–175.
  • 20: Bibliography M
  • A. J. MacLeod (1996a) Algorithm 757: MISCFUN, a software package to compute uncommon special functions. ACM Trans. Math. Software 22 (3), pp. 288–301.
  • K. L. Majumder and G. P. Bhattacharjee (1973) Algorithm AS 63. The incomplete beta integral. Appl. Statist. 22 (3), pp. 409–411.
  • J. W. Miles (1975) Asymptotic approximations for prolate spheroidal wave functions. Studies in Appl. Math. 54 (4), pp. 315–349.
  • C. Mortici (2013b) Further improvements of some double inequalities for bounding the gamma function. Math. Comput. Modelling 57 (5-6), pp. 1360–1363.
  • M. E. Muldoon (1977) Higher monotonicity properties of certain Sturm-Liouville functions. V. Proc. Roy. Soc. Edinburgh Sect. A 77 (1-2), pp. 23–37.