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1: Bibliography F
  • S. Farid Khwaja and A. B. Olde Daalhuis (2014) Uniform asymptotic expansions for hypergeometric functions with large parameters IV. Anal. Appl. (Singap.) 12 (6), pp. 667–710.
  • S. Fempl (1960) Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).
  • C. Ferreira, J. L. López, and E. Pérez Sinusía (2013a) The third Appell function for one large variable. J. Approx. Theory 165, pp. 60–69.
  • C. Ferreira, J. L. López, and E. Pérez Sinusía (2005) Incomplete gamma functions for large values of their variables. Adv. in Appl. Math. 34 (3), pp. 467–485.
  • C. Ferreira, J. L. López, and E. P. Sinusía (2013b) The second Appell function for one large variable. Mediterr. J. Math. 10 (4), pp. 1853–1865.
  • 2: Bibliography B
  • G. Backenstoss (1970) Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.
  • K. L. Bell and N. S. Scott (1980) Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.
  • C. M. Bender and T. T. Wu (1973) Anharmonic oscillator. II. A study of perturbation theory in large order. Phys. Rev. D 7, pp. 1620–1636.
  • W. G. Bickley (1935) Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.
  • L. C. Biedenharn, R. L. Gluckstern, M. H. Hull, and G. Breit (1955) Coulomb functions for large charges and small velocities. Phys. Rev. (2) 97 (2), pp. 542–554.
  • 3: 9.18 Tables
    §9.18(ii) Real Variables
  • Miller (1946) tabulates Ai ( x ) , Ai ( x ) for x = 20 ( .01 ) 2 ; log 10 Ai ( x ) , Ai ( x ) / Ai ( x ) for x = 0 ( .1 ) 25 ( 1 ) 75 ; Bi ( x ) , Bi ( x ) for x = 10 ( .1 ) 2.5 ; log 10 Bi ( x ) , Bi ( x ) / Bi ( x ) for x = 0 ( .1 ) 10 ; M ( x ) , N ( x ) , θ ( x ) , ϕ ( x ) (respectively F ( x ) , G ( x ) , χ ( x ) , ψ ( x ) ) for x = 80 ( 1 ) 30 ( .1 ) 0 . Precision is generally 8D; slightly less for some of the auxiliary functions. Extracts from these tables are included in Abramowitz and Stegun (1964, Chapter 10), together with some auxiliary functions for large arguments.

  • Zhang and Jin (1996, p. 337) tabulates Ai ( x ) , Ai ( x ) , Bi ( x ) , Bi ( x ) for x = 0 ( 1 ) 20 to 8S and for x = 20 ( 1 ) 0 to 9D.

  • §9.18(iii) Complex Variables
  • Sherry (1959) tabulates a k , Ai ( a k ) , a k , Ai ( a k ) , k = 1 ( 1 ) 50 ; 20S.

  • 4: Bibliography O
  • A. B. Olde Daalhuis (2003a) Uniform asymptotic expansions for hypergeometric functions with large parameters. I. Analysis and Applications (Singapore) 1 (1), pp. 111–120.
  • A. B. Olde Daalhuis (2003b) Uniform asymptotic expansions for hypergeometric functions with large parameters. II. Analysis and Applications (Singapore) 1 (1), pp. 121–128.
  • A. B. Olde Daalhuis (2010) Uniform asymptotic expansions for hypergeometric functions with large parameters. III. Analysis and Applications (Singapore) 8 (2), pp. 199–210.
  • F. W. J. Olver (1952) Some new asymptotic expansions for Bessel functions of large orders. Proc. Cambridge Philos. Soc. 48 (3), pp. 414–427.
  • F. W. J. Olver (1962) Tables for Bessel Functions of Moderate or Large Orders. National Physical Laboratory Mathematical Tables, Vol. 6. Department of Scientific and Industrial Research, Her Majesty’s Stationery Office, London.
  • 5: 9.9 Zeros
    If k is regarded as a continuous variable, then … For large k
    9.9.6 a k = T ( 3 8 π ( 4 k 1 ) ) ,
    9.9.7 Ai ( a k ) = ( 1 ) k 1 V ( 3 8 π ( 4 k 1 ) ) ,
    9.9.8 a k = U ( 3 8 π ( 4 k 3 ) ) ,
    6: 28.35 Tables
    §28.35(i) Real Variables
  • Ince (1932) includes eigenvalues a n , b n , and Fourier coefficients for n = 0 or 1 ( 1 ) 6 , q = 0 ( 1 ) 10 ( 2 ) 20 ( 4 ) 40 ; 7D. Also ce n ( x , q ) , se n ( x , q ) for q = 0 ( 1 ) 10 , x = 1 ( 1 ) 90 , corresponding to the eigenvalues in the tables; 5D. Notation: a n = 𝑏𝑒 n 2 q , b n = 𝑏𝑜 n 2 q .

  • Kirkpatrick (1960) contains tables of the modified functions Ce n ( x , q ) , Se n + 1 ( x , q ) for n = 0 ( 1 ) 5 , q = 1 ( 1 ) 20 , x = 0.1 ( .1 ) 1 ; 4D or 5D.

  • National Bureau of Standards (1967) includes the eigenvalues a n ( q ) , b n ( q ) for n = 0 ( 1 ) 3 with q = 0 ( .2 ) 20 ( .5 ) 37 ( 1 ) 100 , and n = 4 ( 1 ) 15 with q = 0 ( 2 ) 100 ; Fourier coefficients for ce n ( x , q ) and se n ( x , q ) for n = 0 ( 1 ) 15 , n = 1 ( 1 ) 15 , respectively, and various values of q in the interval [ 0 , 100 ] ; joining factors g e , n ( q ) , f e , n ( q ) for n = 0 ( 1 ) 15 with q = 0 ( .5  to  10 ) 100 (but in a different notation). Also, eigenvalues for large values of q . Precision is generally 8D.

  • §28.35(ii) Complex Variables
    7: 9.7 Asymptotic Expansions
    For large x , … Numerical values of χ ( n ) are given in Table 9.7.1 for n = 1 ( 1 ) 20 to 2D. …
    §9.7(iii) Error Bounds for Real Variables
    §9.7(iv) Error Bounds for Complex Variables
    provided that n 0 , σ = 1 6 for (9.7.5) and n 1 , σ = 0 for (9.7.6). …
    8: Bibliography L
  • R. E. Langer (1934) The solutions of the Mathieu equation with a complex variable and at least one parameter large. Trans. Amer. Math. Soc. 36 (3), pp. 637–695.
  • H. T. Lau (1995) A Numerical Library in C for Scientists and Engineers. CRC Press, Boca Raton, FL.
  • H. T. Lau (2004) A Numerical Library in Java for Scientists & Engineers. Chapman & Hall/CRC, Boca Raton, FL.
  • D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.
  • J. L. López (1999) Asymptotic expansions of the Whittaker functions for large order parameter. Methods Appl. Anal. 6 (2), pp. 249–256.
  • 9: 28.8 Asymptotic Expansions for Large q
    §28.8 Asymptotic Expansions for Large q
    §28.8(ii) Sips’ Expansions
    §28.8(iii) Goldstein’s Expansions
    Barrett’s Expansions
    10: 30.9 Asymptotic Approximations and Expansions
    §30.9(i) Prolate Spheroidal Wave Functions
    For uniform asymptotic expansions in terms of Airy or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1986). … The cases of large m , and of large m and large | γ | , are studied in Abramowitz (1949). …The behavior of λ n m ( γ 2 ) for complex γ 2 and large | λ n m ( γ 2 ) | is investigated in Hunter and Guerrieri (1982).