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asymptotic approximations for large parameters

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1: 13.8 Asymptotic Approximations for Large Parameters
§13.8 Asymptotic Approximations for Large Parameters
§13.8(i) Large | b | , Fixed a and z
§13.8(ii) Large b and z , Fixed a and b / z
§13.8(iii) Large a
2: 28.26 Asymptotic Approximations for Large q
§28.26 Asymptotic Approximations for Large q
§28.26(ii) Uniform Approximations
3: 28.8 Asymptotic Expansions for Large q
§28.8 Asymptotic Expansions for Large q
Barrett’s Expansions
Dunster’s Approximations
4: 13.20 Uniform Asymptotic Approximations for Large μ
§13.20(i) Large μ , Fixed κ
5: 13.21 Uniform Asymptotic Approximations for Large κ
§13.21 Uniform Asymptotic Approximations for Large κ
6: 8.12 Uniform Asymptotic Expansions for Large Parameter
§8.12 Uniform Asymptotic Expansions for Large Parameter
Inverse Function
7: 2.4 Contour Integrals
§2.4(i) Watson’s Lemma
Then … For examples see Olver (1997b, pp. 315–320).
§2.4(iii) Laplace’s Method
§2.4(v) Coalescing Saddle Points: Chester, Friedman, and Ursell’s Method
8: 2.3 Integrals of a Real Variable
Then … For the Fourier integral … Then the series obtained by substituting (2.3.7) into (2.3.1) and integrating formally term by term yields an asymptotic expansion: … (In other words, differentiation of (2.3.8) with respect to the parameter λ (or μ ) is legitimate.) …
§2.3(iv) Method of Stationary Phase
9: 8.11 Asymptotic Approximations and Expansions
§8.11 Asymptotic Approximations and Expansions
10: 13.22 Zeros
Asymptotic approximations to the zeros when the parameters κ and/or μ are large can be found by reversion of the uniform approximations provided in §§13.20 and 13.21. …