approximations for large parameters

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1: 34.8 Approximations for Large Parameters

§34.8 Approximations for Large Parameters

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2: 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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3: 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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4: 28.8 Asymptotic Expansions for Large $q$

§28.8 Asymptotic Expansions for Large $q$

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Barrett’s Expansions

… ►The approximations apply when the parameters $a$ and $q$ are real and large, and are uniform with respect to various regions in the $z$-plane. … ►

Dunster’s Approximations

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5: 13.20 Uniform Asymptotic Approximations for Large $\mu$

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§13.20(i) Large $\mu$, Fixed $\kappa$

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6: 13.21 Uniform Asymptotic Approximations for Large $\kappa$

§13.21 Uniform Asymptotic Approximations for Large $\kappa$

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7: Bibliography O

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Whittaker functions with both parameters large: Uniform approximations in terms of parabolic cylinder functions. Proc. Roy. Soc. Edinburgh Sect. A 86 (3-4), pp. 213–234.

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External Links:

ISSN 0308-2105, MathReview (P. F. Hsieh), ZentralBlatt

Referenced by:

§13.20(iii), §13.20(iv).

Encodings:

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8: 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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9: 2.4 Contour Integrals

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

… ►in which $z$ is a large real or complex parameter, $p(\alpha ,t)$ and $q(\alpha ,t)$ are analytic functions of $t$ and continuous in $t$ and a second parameter $\alpha$. … ►For large $|z|$, $I(\alpha ,z)$ is approximated uniformly by the integral that corresponds to (2.4.19) when $f(\alpha ,w)$ is replaced by a constant. … ►

§2.4(vi) Other Coalescing Critical Points

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10: 2.3 Integrals of a Real Variable

… ►Then … ►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.) … ►When $p(t)$ is real and $x$ is a large positive parameter, the main contribution to the integral … ►When the parameter $x$ is large the contributions from the real and imaginary parts of the integrand in …