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asymptotic expansions for small parameters

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11: 12.14 The Function W ( a , x )
§12.14(viii) Asymptotic Expansions for Large Variable
§12.14(ix) Uniform Asymptotic Expansions for Large Parameter
Positive a , 2 a < x <
Airy-type Uniform Expansions
12: 14.32 Methods of Computation
In particular, for small or moderate values of the parameters μ and ν the power-series expansions of the various hypergeometric function representations given in §§14.3(i)14.3(iii), 14.19(ii), and 14.20(i) can be selected in such a way that convergence is stable, and reasonably rapid, especially when the argument of the functions is real. …
  • Application of the uniform asymptotic expansions for large values of the parameters given in §§14.15 and 14.20(vii)14.20(ix).

  • 13: 28.15 Expansions for Small q
    §28.15 Expansions for Small q
    §28.15(i) Eigenvalues λ ν ( q )
    28.15.1 λ ν ( q ) = ν 2 + 1 2 ( ν 2 1 ) q 2 + 5 ν 2 + 7 32 ( ν 2 1 ) 3 ( ν 2 4 ) q 4 + 9 ν 4 + 58 ν 2 + 29 64 ( ν 2 1 ) 5 ( ν 2 4 ) ( ν 2 9 ) q 6 + .
    28.15.2 a ν 2 q 2 a ( ν + 2 ) 2 q 2 a ( ν + 4 ) 2 = q 2 a ( ν 2 ) 2 q 2 a ( ν 4 ) 2 .
    28.15.3 me ν ( z , q ) = e i ν z q 4 ( 1 ν + 1 e i ( ν + 2 ) z 1 ν 1 e i ( ν 2 ) z ) + q 2 32 ( 1 ( ν + 1 ) ( ν + 2 ) e i ( ν + 4 ) z + 1 ( ν 1 ) ( ν 2 ) e i ( ν 4 ) z 2 ( ν 2 + 1 ) ( ν 2 1 ) 2 e i ν z ) + ;
    14: 10.41 Asymptotic Expansions for Large Order
    §10.41 Asymptotic Expansions for Large Order
    §10.41(i) Asymptotic Forms
    §10.41(iv) Double Asymptotic Properties
    §10.41(v) Double Asymptotic Properties (Continued)
    15: 12.10 Uniform Asymptotic Expansions for Large Parameter
    §12.10 Uniform Asymptotic Expansions for Large Parameter
    §12.10(vi) Modifications of Expansions in Elementary Functions
    Modified Expansions
    16: 10.69 Uniform Asymptotic Expansions for Large Order
    §10.69 Uniform Asymptotic Expansions for Large Order
    All fractional powers take their principal values. All four expansions also enjoy the same kind of double asymptotic property described in §10.41(iv). …
    17: 10.68 Modulus and Phase Functions
    §10.68(iii) Asymptotic Expansions for Large Argument
    10.68.16 M ν ( x ) = e x / 2 ( 2 π x ) 1 2 ( 1 μ 1 8 2 1 x + ( μ 1 ) 2 256 1 x 2 ( μ 1 ) ( μ 2 + 14 μ 399 ) 6144 2 1 x 3 + O ( 1 x 4 ) ) ,
    10.68.17 ln M ν ( x ) = x 2 1 2 ln ( 2 π x ) μ 1 8 2 1 x ( μ 1 ) ( μ 25 ) 384 2 1 x 3 ( μ 1 ) ( μ 13 ) 128 1 x 4 + O ( 1 x 5 ) ,
    18: 10.1 Special Notation
    m , n integers. In §§10.4710.71 n is nonnegative.
    ν real or complex parameter (the order).
    δ arbitrary small positive constant.
    For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).
    19: 10.67 Asymptotic Expansions for Large Argument
    §10.67 Asymptotic Expansions for Large Argument
    §10.67(i) ber ν x , bei ν x , ker ν x , kei ν x , and Derivatives
    The contributions of the terms in ker ν x , kei ν x , ker ν x , and kei ν x on the right-hand sides of (10.67.3), (10.67.4), (10.67.7), and (10.67.8) are exponentially small compared with the other terms, and hence can be neglected in the sense of Poincaré asymptotic expansions2.1(iii)). …
    §10.67(ii) Cross-Products and Sums of Squares in the Case ν = 0
    20: 15.12 Asymptotic Approximations
    §15.12 Asymptotic Approximations
    Let δ denote an arbitrary small positive constant. … For this result and an extension to an asymptotic expansion with error bounds see Jones (2001). … For U ( a , z ) see §12.2, and for an extension to an asymptotic expansion see Olde Daalhuis (2003a). … For Ai ( z ) see §9.2, and for further information and an extension to an asymptotic expansion see Olde Daalhuis (2003b). …