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modified expansions in terms of elementary functions

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1: 12.10 Uniform Asymptotic Expansions for Large Parameter
§12.10(vi) Modifications of Expansions in Elementary Functions
2: 28.8 Asymptotic Expansions for Large q
§28.8(ii) Sips’ Expansions
Barrett’s Expansions
The approximants are elementary functions, Airy functions, Bessel functions, and parabolic cylinder functions; compare §2.8. … With additional restrictions on z , uniform asymptotic approximations for solutions of (28.2.1) and (28.20.1) are also obtained in terms of elementary functions by re-expansions of the Whittaker functions; compare §2.8(ii). Subsequently the asymptotic solutions involving either elementary or Whittaker functions are identified in terms of the Floquet solutions me ν ( z , q ) 28.12(ii)) and modified Mathieu functions M ν ( j ) ( z , h ) 28.20(iii)). …
3: 18.15 Asymptotic Approximations
These expansions are in terms of Whittaker functions13.14). … These expansions are in terms of Bessel functions and modified Bessel functions, respectively. …
In Terms of Elementary Functions
In Terms of Bessel Functions
In Terms of Airy Functions
4: 2.8 Differential Equations with a Parameter
These are elementary functions in Case I, and Airy functions9.2) in Case II. … For other examples of uniform asymptotic approximations and expansions of special functions in terms of Airy functions see especially §10.20 and §§12.10(vii), 12.10(viii); also §§12.14(ix), 13.20(v), 13.21(iii), 13.21(iv), 15.12(iii), 18.15(iv), 30.9(i), 30.9(ii), 32.11(ii), 32.11(iii), 33.12(i), 33.12(ii), 33.20(iv), 36.12(ii), 36.13. … For other examples of uniform asymptotic approximations and expansions of special functions in terms of Bessel functions or modified Bessel functions of fixed order see §§13.8(iii), 13.21(i), 13.21(iv), 14.15(i), 14.15(iii), 14.20(vii), 15.12(iii), 18.15(i), 18.15(iv), 18.24, 33.20(iv). … For further examples of uniform asymptotic approximations in terms of parabolic cylinder functions see §§13.20(iii), 13.20(iv), 14.15(v), 15.12(iii), 18.24. For further examples of uniform asymptotic approximations in terms of Bessel functions or modified Bessel functions of variable order see §§13.21(ii), 14.15(ii), 14.15(iv), 14.20(viii), 30.9(i), 30.9(ii). …
5: 33.20 Expansions for Small | ϵ |
§33.20(i) Case ϵ = 0
§33.20(ii) Power-Series in ϵ for the Regular Solution
The functions Y and K are as in §§10.2(ii), 10.25(ii), and the coefficients C k , p are given by (33.20.6).
§33.20(iv) Uniform Asymptotic Expansions
These expansions are in terms of elementary functions, Airy functions, and Bessel functions of orders 2 + 1 and 2 + 2 .
6: 14.15 Uniform Asymptotic Approximations
In other words, the convergent hypergeometric series expansions of P ν - μ ( ± x ) are also generalized (and uniform) asymptotic expansions as μ , with scale 1 / Γ ( j + 1 + μ ) , j = 0 , 1 , 2 , ; compare §2.1(v). … Here I and K are the modified Bessel functions10.25(ii)). … For asymptotic expansions and explicit error bounds, see Dunster (2003b). … For convergent series expansions see Dunster (2004). … See also Olver (1997b, pp. 311–313) and §18.15(iii) for a generalized asymptotic expansion in terms of elementary functions for Legendre polynomials P n ( cos θ ) as n with θ fixed. …
7: 3.10 Continued Fractions
if the expansion of its n th convergent C n in ascending powers of z agrees with (3.10.7) up to and including the term in z n - 1 , n = 1 , 2 , 3 , . … We say that it is associated with the formal power series f ( z ) in (3.10.7) if the expansion of its n th convergent C n in ascending powers of z , agrees with (3.10.7) up to and including the term in z 2 n - 1 , n = 1 , 2 , 3 , . … For elementary functions, see §§ 4.9 and 4.35. … The A n and B n of (3.10.2) can be computed by means of three-term recurrence relations (1.12.5). … In Gautschi (1979c) the forward series algorithm is used for the evaluation of a continued fraction of an incomplete gamma function (see §8.9). …
8: Bibliography C
  • A. Cayley (1895) An Elementary Treatise on Elliptic Functions. George Bell and Sons, London.
  • A. Cayley (1961) An Elementary Treatise on Elliptic Functions. Dover Publications, New York (English).
  • T. M. Cherry (1948) Expansions in terms of parabolic cylinder functions. Proc. Edinburgh Math. Soc. (2) 8, pp. 50–65.
  • W. J. Cody and W. Waite (1980) Software Manual for the Elementary Functions. Prentice-Hall, Englewood Cliffs.
  • W. J. Cody (1993a) Algorithm 714: CELEFUNT – A portable test package for complex elementary functions. ACM Trans. Math. Software 19 (1), pp. 1–21.
  • 9: Bibliography L
  • M. Yu. Loenko (2001) Evaluating elementary functions with guaranteed precision. Programming and Computer Software 27 (2), pp. 101–110.
  • J. L. López and N. M. Temme (1999c) Uniform approximations of Bernoulli and Euler polynomials in terms of hyperbolic functions. Stud. Appl. Math. 103 (3), pp. 241–258.
  • L. Lorch (1990) Monotonicity in terms of order of the zeros of the derivatives of Bessel functions. Proc. Amer. Math. Soc. 108 (2), pp. 387–389.
  • T. A. Lowdon (1970) Integral representation of the Hankel function in terms of parabolic cylinder functions. Quart. J. Mech. Appl. Math. 23 (3), pp. 315–327.
  • W. Luther (1995) Highly accurate tables for elementary functions. BIT 35 (3), pp. 352–360.
  • 10: Bibliography S
  • H. Shanker (1939) On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 3, pp. 226–230.
  • A. Sharples (1971) Uniform asymptotic expansions of modified Mathieu functions. J. Reine Angew. Math. 247, pp. 1–17.
  • T. Shiota (1986) Characterization of Jacobian varieties in terms of soliton equations. Invent. Math. 83 (2), pp. 333–382.
  • D. M. Smith (1989) Efficient multiple-precision evaluation of elementary functions. Math. Comp. 52 (185), pp. 131–134.
  • K. Soni (1980) Exact error terms in the asymptotic expansion of a class of integral transforms. I. Oscillatory kernels. SIAM J. Math. Anal. 11 (5), pp. 828–841.