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1: 16.26 Approximations
For discussions of the approximation of generalized hypergeometric functions and the Meijer G -function in terms of polynomials, rational functions, and Chebyshev polynomials see Luke (1975, §§5.12 - 5.13) and Luke (1977b, Chapters 1 and 9).
2: Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)
Such a solution is given in terms of a Riemann theta function with two phases. …
3: 19.38 Approximations
Minimax polynomial approximations (§3.11(i)) for K ( k ) and E ( k ) in terms of m = k 2 with 0 m < 1 can be found in Abramowitz and Stegun (1964, §17.3) with maximum absolute errors ranging from 4×10⁻⁵ to 2×10⁻⁸. … The accuracy is controlled by the number of terms retained in the approximation; for real variables the number of significant figures appears to be roughly twice the number of terms retained, perhaps even for ϕ near π / 2 with the improvements made in the 1970 reference. …
4: 33.19 Power-Series Expansions in r
33.19.1 f ( ϵ , ; r ) = r + 1 k = 0 α k r k ,
Here κ is defined by (33.14.6), A ( ϵ , ) is defined by (33.14.11) or (33.14.12), γ 0 = 1 , γ 1 = 1 , and …
33.19.6 k ( k + 2 + 1 ) δ k + 2 δ k 1 + ϵ δ k 2 + 2 ( 2 k + 2 + 1 ) A ( ϵ , ) α k = 0 , k = 2 , 3 , ,
with β 0 = β 1 = 0 , and
33.19.7 β k β k 1 + 1 4 ( k 1 ) ( k 2 2 ) ϵ β k 2 + 1 2 ( k 1 ) ϵ γ k 2 = 0 , k = 2 , 3 , .
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  • 6: 12.16 Mathematical Applications
    7: 18.24 Hahn Class: Asymptotic Approximations
    In particular, asymptotic formulas in terms of elementary functions are given when z = x is real and fixed. … This expansion is in terms of the parabolic cylinder function and its derivative. … This expansion is in terms of confluent hypergeometric functions. … Both expansions are in terms of parabolic cylinder functions. …
    Approximations in Terms of Laguerre Polynomials
    8: 6.12 Asymptotic Expansions
    For these and other error bounds see Olver (1997b, pp. 109–112) with α = 0 . For re-expansions of the remainder term leading to larger sectors of validity, exponential improvement, and a smooth interpretation of the Stokes phenomenon, see §§2.11(ii)2.11(iv), with p = 1 . …If the expansion is terminated at the n th term, then the remainder term is bounded by 1 + χ ( n + 1 ) times the next term. … The remainder terms are given by …When | ph z | 1 4 π , these remainders are bounded in magnitude by the first neglected terms in (6.12.3) and (6.12.4), respectively, and have the same signs as these terms when ph z = 0 . …
    9: 7.16 Generalized Error Functions
    These functions can be expressed in terms of the incomplete gamma function γ ( a , z ) 8.2(i)) by change of integration variable.
    10: 16.7 Relations to Other Functions
    Further representations of special functions in terms of F q p functions are given in Luke (1969a, §§6.2–6.3), and an extensive list of F q q + 1 functions with rational numbers as parameters is given in Krupnikov and Kölbig (1997).