asymptotic approximations and expansions for large |r|

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1: 33.21 Asymptotic Approximations for Large $|r|$

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§33.21(i) Limiting Forms

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§33.21(ii) Asymptotic Expansions

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2: 2.11 Remainder Terms; Stokes Phenomenon

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§2.11(i) Numerical Use of Asymptotic Expansions

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§2.11(iii) Exponentially-Improved Expansions

… ►For another approach see Paris (2001a, b). ►

§2.11(vi) Direct Numerical Transformations

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3: 2.10 Sums and Sequences

§2.10 Sums and Sequences

… ► … ►This identity can be used to find asymptotic approximations for large $n$ when the factor $v_j$ changes slowly with $j$, and $u_j$ is oscillatory; compare the approximation of Fourier integrals by integration by parts in §2.3(i). … ►

§2.10(iii) Asymptotic Expansions of Entire Functions

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The coefficients in the Laurent expansion

2.10.27 $$g(z)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{g}_n{z}^n,$$ ,

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Symbols:

$r$: radious of annulus, $g(z)$: comparison function and $g_n$: coefficients

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http://dlmf.nist.gov/2.10.E27

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Annotations for §2.10(iv), §2.10 and Ch.2

have known asymptotic behavior as $n\to \pm \mathrm{\infty }$.

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4: 8.11 Asymptotic Approximations and Expansions

§8.11 Asymptotic Approximations and Expansions

… ►where $\delta$ denotes an arbitrary small positive constant. … ►For an exponentially-improved asymptotic expansion (§2.11(iii)) see Olver (1991a). … ►This expansion is absolutely convergent for all finite $z$, and it can also be regarded as a generalized asymptotic expansion (§2.1(v)) of $\gamma (a,z)$ as $a\to \mathrm{\infty }$ in $|\mathrm{ph}a|\le \pi -\delta$. … ►

5: Bibliography W

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An asymptotic expansion of $W_{k,m}(z)$ with large variable and parameters. Math. Comp. 27 (122), pp. 429–436.

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External Links:

FullText, Document, MathReview (J. A. Donaldson), ZentralBlatt

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§13.19.

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On uniform asymptotic expansion of definite integrals. J. Approximation Theory 7 (1), pp. 76–86.

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External Links:

Document, MathReview (L. Berg), ZentralBlatt

Referenced by:

§8.11(v).

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Asymptotic expansions of the Kontorovich-Lebedev transform. Appl. Anal. 12 (3), pp. 161–172.

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External Links:

ISSN 0003-6811, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

§10.43(v).

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Applications of some recent results in asymptotic expansions. Congr. Numer. 37, pp. 145–182.

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External Links:

ISSN 0384-9864, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§19.27(vi).

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Error bounds for asymptotic approximations of special functions. Ann. Numer. Math. 2 (1-4), pp. 181–197.

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Notes:

Special functions (Torino, 1993)

External Links:

ISSN 1021-2655, MathReview (Andrei Martínez Finkelshtein), ZentralBlatt

Referenced by:

§10.21(vi).

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6: 2.6 Distributional Methods

§2.6 Distributional Methods

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§2.6(ii) Stieltjes Transform

… ►To derive an asymptotic expansion of $𝒮f\left(z\right)$ for large values of $|z|$, with , we assume that $f(t)$ possesses an asymptotic expansion of the form … ►An application has been given by López (2000) to derive asymptotic expansions of standard symmetric elliptic integrals, complete with error bounds; see §19.27(vi). … ►We now derive an asymptotic expansion of ${𝐼}^{\mu }f(x)$ for large positive values of $x$. …

7: 13.9 Zeros

… ► … ►where $n$ is a large positive integer. … ►where $n$ is a large positive integer. ►For fixed $a$ and $z$ in $ℂ$, $U(a,b,z)$ has two infinite strings of $b$-zeros that are asymptotic to the imaginary axis as $|b|\to \mathrm{\infty }$.

8: 18.26 Wilson Class: Continued

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18.26.9 $$\underset{\beta \to \mathrm{\infty }}{lim}{R}_n(x;-N-1,\beta ,\gamma ,\delta )=R_n(x;\gamma ,\delta ,N).$$

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Symbols:

$R_n(x;\alpha ,\beta ,\gamma ,\delta )$: Racah polynomial, $R_n(x;\gamma ,\delta ,N)$: dual Hahn polynomial, $N$: positive integer, $\delta$: arbitrary small positive constant, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.26(ii)

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http://dlmf.nist.gov/18.26.E9

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Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

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18.26.12 $$r(x;\beta ,c,N)=x(x+\beta +c^{-1}(1-c)N),$$

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Defines:

$r(x;\beta ,c,N)$ (locally)

Symbols:

$N$: positive integer and $x$: real variable

Referenced by:

§18.26(ii)

Permalink:

http://dlmf.nist.gov/18.26.E12

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Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

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18.26.13 $$\underset{N\to \mathrm{\infty }}{lim}{R}_n(r(x;\beta ,c,N);\beta -1,c^{-1}(1-c)N,N)=M_n(x;\beta ,c).$$

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Symbols:

$M_n(x;\beta ,c)$: Meixner polynomial, $R_n(x;\gamma ,\delta ,N)$: dual Hahn polynomial, $N$: positive integer, $n$: nonnegative integer, $x$: real variable and $r(x;\beta ,c,N)$

Referenced by:

§18.26(ii)

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http://dlmf.nist.gov/18.26.E13

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Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

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§18.26(v) Asymptotic Approximations

►For asymptotic expansions of Wilson polynomials of large degree see Wilson (1991), and for asymptotic approximations to their largest zeros see Chen and Ismail (1998). …

9: 2.7 Differential Equations

… ►Note that the coefficients in the expansions (2.7.12), (2.7.13) for the “late” coefficients, that is, $a_{s,1}$, $a_{s,2}$ with $s$ large, are the “early” coefficients $a_{j,2}$, $a_{j,1}$ with $j$ small. … ►

§2.7(iii) Liouville–Green (WKBJ) Approximation

►For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows: … ►By approximating … ►The first of these references includes extensions to complex variables and reversions for zeros. …

10: Bibliography B

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Chapter 13 of Ramanujan’s second notebook: Integrals and asymptotic expansions. Expo. Math. 2 (4), pp. 289–347.

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External Links:

ISSN 0723-0869, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§2.10(iii).

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Asymptotic Expansions of Integrals. Holt, Rinehart, and Winston, New York.

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Notes:

Reprinted with corrections by Dover Publications Inc., New York, 1986

External Links:

ISBN 0-486-65082-0, MathReview, ZentralBlatt

Referenced by:

§2.3(ii), §2.3(v), §2.4(iv), §2.4(v), §2.5(iii), §2.5(ii), Ch.2, Ch.9.

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Uniform asymptotic expansion of Charlier polynomials. Methods Appl. Anal. 1 (3), pp. 294–313.

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External Links:

ISSN 1073-2772, Pdf, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§18.24.

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Problem of two Coulomb centres at large intercentre separation: Asymptotic expansions from analytical solutions of the Heun equation. J. Phys. A 30 (2), pp. 559–571.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§31.13.

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Approximations for the late coefficients in asymptotic expansions arising in the method of steepest descents. Methods Appl. Anal. 2 (4), pp. 475–489.

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External Links:

ISSN 1073-2772, Pdf, MathReview (Dan Polis̆evski), ZentralBlatt

Referenced by:

§2.4(iv).

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