asymptotic expansion for large argument

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31: 30.11 Radial Spheroidal Wave Functions

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§30.11(iii) Asymptotic Behavior

… ►For asymptotic expansions in negative powers of $z$ see Meixner and Schäfke (1954, p. 293). … ►

30.11.8 $$S_n^{m(1)}(z,\gamma )=K_n^m(\gamma ){\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2),$$

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Symbols:

$S_n^{m(j)}(z,\gamma )$: radial spheroidal wave function, ${\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2)$: spheroidal wave function of complex argument, $z$: complex variable, $m$: nonnegative integer, $n\ge m$: integer degree, $K_n^m(\gamma )$: connection coefficient and ${\gamma }^2$: real parameter

A&S Ref:

21.10.1 (in different form)

Permalink:

http://dlmf.nist.gov/30.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §30.11(v), §30.11 and Ch.30

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30.11.9 $$S_n^{m(2)}(z,\gamma )=\frac{(n-m)!}{(n+m)!}\frac{{(-1)}^{m+1}{\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2)}{\gamma K_n^m(\gamma )A_n^m({\gamma }^2)A_n^{-m}({\gamma }^2)},$$

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Symbols:

$!$: factorial (as in $n!$), $S_n^{m(j)}(z,\gamma )$: radial spheroidal wave function, ${\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2)$: spheroidal wave function of complex argument, $z$: complex variable, $m$: nonnegative integer, $n\ge m$: integer degree, $A_n^m({\gamma }^2)$ , $K_n^m(\gamma )$: connection coefficient and ${\gamma }^2$: real parameter

A&S Ref:

21.10.2 (in different form)

Permalink:

http://dlmf.nist.gov/30.11.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §30.11(v), §30.11 and Ch.30

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32: 4.45 Methods of Computation

… ►Another method, when $x$ is large, is to sum … ►The inverses $\mathrm{arcsinh}$, $\mathrm{arccosh}$, and $\mathrm{arctanh}$ can be computed from the logarithmic forms given in §4.37(iv), with real arguments. … ►Initial approximations are obtainable, for example, from the power series (4.13.6) (with $t\ge 0$) when $x$ is close to $-1/\mathrm{e}$, from the asymptotic expansion (4.13.10) when $x$ is large, and by numerical integration of the differential equation (4.13.4) (§3.7) for other values of $x$. …

33: 14.20 Conical (or Mehler) Functions

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§14.20(vii) Asymptotic Approximations: Large $\tau$, Fixed $\mu$

… ►For asymptotic expansions and explicit error bounds, see Olver (1997b, pp. 473–474). … ►

§14.20(viii) Asymptotic Approximations: Large $\tau$, $0\le \mu \le A\tau$

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§14.20(ix) Asymptotic Approximations: Large $\mu$, $0\le \tau \le A\mu$

… ►For extensions to complex arguments (including the range ), asymptotic expansions, and explicit error bounds, see Dunster (1991). …

34: Bibliography S

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Uniform asymptotic expansions of modified Mathieu functions. J. Reine Angew. Math. 247, pp. 1–17.

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External Links:

MathReview (J. Meixner), ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

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Asymptotic expansions of Mellin transforms and analogues of Watson’s lemma. SIAM J. Math. Anal. 16 (4), pp. 896–906.

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External Links:

ISSN 0036-1410, Document, Link, MathReview (John S. Lew), ZentralBlatt

Referenced by:

§2.3(vi).

Encodings:

BibTeX

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A simple approach to asymptotic expansions for Fourier integrals of singular functions. Appl. Math. Comput. 216 (11), pp. 3378–3385.

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External Links:

ISSN 0096-3003, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§2.3(iv).

Encodings:

BibTeX

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Asymptotic expansion of Mellin transforms in the complex plane. Int. J. Pure Appl. Math. 71 (3), pp. 465–480.

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External Links:

ISSN 1311-8080, MathReview Entry, ZentralBlatt

Referenced by:

§2.3(vi).

Encodings:

BibTeX

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Uniform asymptotic expansions of Hermite polynomials. M. Phil. thesis, City University of Hong Kong.

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External Links:

Abstract

Referenced by:

§18.16(v).

Encodings:

BibTeX

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35: Bibliography C

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Asymptotic behaviour of the zeros of the (generalized) Laguerre polynomial $L_n^{\alpha }(x)$ as the index $\alpha \to \mathrm{\infty }$ and limiting formula relating Laguerre polynomials of large index and large argument to Hermite polynomials. Lett. Nuovo Cimento (2) 23 (3), pp. 101–102.

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External Links:

ISSN 0024-1318, Document, MathReview (R. A. Askey)

Referenced by:

§18.7(iii).

Encodings:

BibTeX

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Asymptotic expansion of the first elliptic integral. SIAM J. Math. Anal. 16 (5), pp. 1072–1092.

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External Links:

ISSN 0036-1410, Document, MathReview (Kusum Soni), ZentralBlatt

Referenced by:

§19.27(ii), §19.27(vi), §19.9(ii), §2.5(ii).

Encodings:

BibTeX

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Uniform asymptotic expansion of an integral with a saddle point, a pole and a branch point. Proc. Roy. Soc. London Ser. A 426, pp. 273–286.

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External Links:

ISSN 0080-4630, FullText, MathReview, ZentralBlatt

Referenced by:

§2.4(vi).

Encodings:

BibTeX

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On the asymptotic expansion of Airy’s integral. Proc. Glasgow Math. Assoc. 6, pp. 113–115.

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External Links:

Document, MathReview (L. Berg), ZentralBlatt

Referenced by:

(9.5.7), §9.5(ii).

Encodings:

BibTeX

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Asymptotic Expansions. Cambridge Tracts in Mathematics and Mathematical Physics, Cambridge University Press, New York.

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External Links:

MathReview (A. E. Danese), ZentralBlatt

Referenced by:

Ch.2.

Encodings:

BibTeX

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36: Bibliography B

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Asymptotic expansions of the modified Bessel function of the third kind of imaginary order. SIAM J. Appl. Math. 15, pp. 1315–1323.

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External Links:

ISSN 0036-1399, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.45.

Encodings:

BibTeX

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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).

Encodings:

BibTeX

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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.

Encodings:

BibTeX

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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.

Encodings:

BibTeX

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On the asymptotic expansion of some integrals. Arch. Math. (Basel) 42 (3), pp. 253–259.

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External Links:

ISSN 0003-889X, Document, MathReview (E. Riekstiņš), ZentralBlatt

Referenced by:

§2.5(ii).

Encodings:

BibTeX

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37: Bibliography G

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Exactification of the Poincaré asymptotic expansion of the Hankel integral: spectacularly accurate asymptotic expansions and non-asymptotic scales. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470 (2162), pp. 20130529, 16.

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External Links:

ISSN 1364-5021, Document, Link, MathReview (Eugenia N. Petropoulou), ZentralBlatt

Referenced by:

§10.22(v), §10.22(v).

Encodings:

BibTeX

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Algorithm 259: Legendre functions for arguments larger than one. Comm. ACM 8 (8), pp. 488–492.

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Notes:

Variable-precision Algol. For remark see ACM Trans. Math. Software 3(2) (1977), p. 204–205

External Links:

ISSN 0001-0782, Document

Referenced by:

§14.34(ii).

Encodings:

BibTeX

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Some integrals involving three Bessel functions when their arguments satisfy the triangle inequalities. J. Math. Phys. 25 (11), pp. 3350–3356.

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External Links:

ISSN 0022-2488, Document, MathReview (B. R. Bhonsle), ZentralBlatt

Referenced by:

§10.22(iv), §10.43(iv).

Encodings:

BibTeX

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Computation of the modified Bessel function of the third kind of imaginary orders: Uniform Airy-type asymptotic expansion. J. Comput. Appl. Math. 153 (1-2), pp. 225–234.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§10.45, §10.74(viii).

Encodings:

BibTeX

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Asymptotische Entwicklungen für unvollständige Gammafunktionen. Forum Math. 3 (2), pp. 105–141 (German).

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Notes:

Translated title: Asymptotic expansions for incomplete gamma functions.

External Links:

ISSN 0933-7741, MathReview (T. Erber), ZentralBlatt

Referenced by:

§8.16.

Encodings:

BibTeX

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38: 12.14 The Function $W(a,x)$

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§12.14(viii) Asymptotic Expansions for Large Variable

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§12.14(ix) Uniform Asymptotic Expansions for Large Parameter

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Airy-type Uniform Expansions

… ► … ►For properties of the modulus and phase functions, including differential equations and asymptotic expansions for large $x$, see Miller (1955, pp. 87–88). …