asymptotic expansions for small parameters
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11: 12.14 The Function
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§12.14(viii) Asymptotic Expansions for Large Variable
… ►§12.14(ix) Uniform Asymptotic Expansions for Large Parameter
… ►Positive ,
… ►Airy-type Uniform Expansions
… ► …12: 14.32 Methods of Computation
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►In particular, for small or moderate values of the parameters
and 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.
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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
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
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10.68.1
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10.68.2
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§10.68(iii) Asymptotic Expansions for Large Argument
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10.68.16
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10.68.17
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18: 10.1 Special Notation
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►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).
integers. In §§10.47–10.71 is nonnegative. | |
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real or complex parameter (the order). | |
arbitrary small positive constant. | |
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