asymptotic
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11: 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). … ►12: 8.16 Generalizations
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►For a generalization of the incomplete gamma function, including asymptotic approximations, see Chaudhry and Zubair (1994, 2001) and Chaudhry et al. (1996).
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13: 28.16 Asymptotic Expansions for Large
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: 6.12 Asymptotic Expansions
§6.12 Asymptotic Expansions
►§6.12(i) Exponential and Logarithmic Integrals
… ►For the function see §9.7(i). … ►§6.12(ii) Sine and Cosine Integrals
… ► …16: 10.70 Zeros
17: 36.15 Methods of Computation
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§36.15(ii) Asymptotics
►Far from the bifurcation set, the leading-order asymptotic formulas of §36.11 reproduce accurately the form of the function, including the geometry of the zeros described in §36.7. Close to the bifurcation set but far from , the uniform asymptotic approximations of §36.12 can be used. … ►Direct numerical evaluation can be carried out along a contour that runs along the segment of the real -axis containing all real critical points of and is deformed outside this range so as to reach infinity along the asymptotic valleys of . … ►This can be carried out by direct numerical evaluation of canonical integrals along a finite segment of the real axis including all real critical points of , with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. …18: 33.21 Asymptotic Approximations for Large
§33.21 Asymptotic Approximations for Large
►§33.21(i) Limiting Forms
… ►§33.21(ii) Asymptotic Expansions
►For asymptotic expansions of and as with and fixed, see Curtis (1964a, §6).19: 2 Asymptotic Approximations
Chapter 2 Asymptotic Approximations
…20: 16.25 Methods of Computation
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►Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations.
…This occurs when the wanted solution is intermediate in asymptotic growth compared with other solutions.
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