uniform expansions

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1: 14.26 Uniform Asymptotic Expansions

§14.26 Uniform Asymptotic Expansions

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2: 9.15 Mathematical Applications

… ►Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …

3: 33.12 Asymptotic Expansions for Large $\eta$

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§33.12(ii) Uniform Expansions

… ►The first set is in terms of Airy functions and the expansions are uniform for fixed $\mathrm{\ell }$ and , where $\delta$ is an arbitrary small positive constant. … ►

4: 10.69 Uniform Asymptotic Expansions for Large Order

§10.69 Uniform Asymptotic Expansions for Large Order

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5: 10.57 Uniform Asymptotic Expansions for Large Order

§10.57 Uniform Asymptotic Expansions for Large Order

►Asymptotic expansions for ${𝗃}_n\left((n+\frac{1}{2})z\right)$, ${𝗒}_n\left((n+\frac{1}{2})z\right)$, ${𝗁}_n^{(1)}\left((n+\frac{1}{2})z\right)$, ${𝗁}_n^{(2)}\left((n+\frac{1}{2})z\right)$, ${𝗂}_n^{(1)}\left((n+\frac{1}{2})z\right)$, and ${𝗄}_n\left((n+\frac{1}{2})z\right)$ as $n\to \mathrm{\infty }$ that are uniform with respect to $z$ can be obtained from the results given in §§10.20 and 10.41 by use of the definitions (10.47.3)–(10.47.7) and (10.47.9). …

6: 8.25 Methods of Computation

… ►DiDonato and Morris (1986) describes an algorithm for computing $P(a,x)$ and $Q(a,x)$ for $a\ge 0$, $x\ge 0$, and $a+x\ne 0$ from the uniform expansions in §8.12. …

7: Bibliography Q

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Uniform asymptotic expansions of a double integral: Coalescence of two stationary points. Proc. Roy. Soc. London Ser. A 456, pp. 407–431.

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External Links:

ISSN 1364-5021, Document, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§2.4(vi).

Encodings:

BibTeX

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8: 10.41 Asymptotic Expansions for Large Order

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§10.41(ii) Uniform Expansions for Real Variable

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10.41.4 $$K_{\nu }\left(\nu z\right)\sim {\left(\frac{\pi }{2\nu }\right)}^{\frac{1}{2}}\frac{{\mathrm{e}}^{-\nu \eta }}{{(1+z^2)}^{\frac{1}{4}}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{U_k(p)}{{\nu }^k},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\eta$ and $U_k(p)$: polynomial coefficient

A&S Ref:

9.7.8

Permalink:

http://dlmf.nist.gov/10.41.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.41(ii), §10.41 and Ch.10

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§10.41(iii) Uniform Expansions for Complex Variable

… ► … ►Similar analysis can be developed for the uniform asymptotic expansions in terms of Airy functions given in §10.20. …

9: 8.12 Uniform Asymptotic Expansions for Large Parameter

§8.12 Uniform Asymptotic Expansions for Large Parameter

… ►A different type of uniform expansion with coefficients that do not possess a removable singularity at $z=a$ is given by … ►

Inverse Function

… ►These expansions involve the inverse error function $\mathrm{inverfc}\left(x\right)$ (§7.17), and are uniform with respect to $q\in [0,1]$. …

10: 14.32 Methods of Computation

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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).

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