asymptotic approximations for large parameters

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1: 13.8 Asymptotic Approximations for Large Parameters

§13.8 Asymptotic Approximations for Large Parameters

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§13.8(ii) Large $b$ and $z$, Fixed $a$ and $b/z$

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§13.8(iii) Large $a$

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§13.8(iv) Large $a$ and $b$

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2: 28.26 Asymptotic Approximations for Large $q$

§28.26 Asymptotic Approximations for Large $q$

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§28.26(ii) Uniform Approximations

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3: 28.8 Asymptotic Expansions for Large $q$

§28.8 Asymptotic Expansions for Large $q$

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Barrett’s Expansions

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Dunster’s Approximations

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4: 13.20 Uniform Asymptotic Approximations for Large $\mu$

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§13.20(i) Large $\mu$, Fixed $\kappa$

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5: 13.21 Uniform Asymptotic Approximations for Large $\kappa$

§13.21 Uniform Asymptotic Approximations for Large $\kappa$

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6: 8.12 Uniform Asymptotic Expansions for Large Parameter

§8.12 Uniform Asymptotic Expansions for Large Parameter

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Inverse Function

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7: 2.4 Contour Integrals

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§2.4(i) Watson’s Lemma

… ►Then … ►For examples see Olver (1997b, pp. 315–320). ►

§2.4(iii) Laplace’s Method

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§2.4(v) Coalescing Saddle Points: Chester, Friedman, and Ursell’s Method

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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 $\lambda$ (or $\mu$) is legitimate.) … ►

§2.3(vi) Asymptotics of Mellin Transforms

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9: 8.11 Asymptotic Approximations and Expansions

§8.11 Asymptotic Approximations and Expansions

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10: 13.22 Zeros

… ►Asymptotic approximations to the zeros when the parameters $\kappa$ and/or $\mu$ are large can be found by reversion of the uniform approximations provided in §§13.20 and 13.21. …