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1: 30.9 Asymptotic Approximations and Expansions

§30.9 Asymptotic Approximations and Expansions

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§30.9(i) Prolate Spheroidal Wave Functions

… ►The asymptotic behavior of ${\lambda }_n^m\left({\gamma }^2\right)$ and $a_{n,k}^m({\gamma }^2)$ as $n\to \mathrm{\infty }$ in descending powers of $2n+1$ is derived in Meixner (1944). …The asymptotic behavior of ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ and ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$ as $x\to \pm 1$ is given in Erdélyi et al. (1955, p. 151). …

2: Bibliography N

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On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

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External Links:

ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§28.26(ii).

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On an asymptotic expansion of the Kontorovich-Lebedev transform. Applicable Anal. 39 (4), pp. 249–263.

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ISSN 0003-6811, Document, MathReview (Aloknath Chakrabarti), ZentralBlatt

Referenced by:

§10.43(v).

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Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

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External Links:

FullText, MathReview (I. A. Stegun), ZentralBlatt

Referenced by:

§19.37(iv).

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A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.

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Notes:

See for additions and corrections same journal, Sect. B 75, 149–163 (1975)

External Links:

MathReview, ZentralBlatt

Referenced by:

§7.14(iii), §7.21, §7.7(iii).

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The asymptotic behavior of the general real solution of the third Painlevé equation. Dokl. Akad. Nauk SSSR 283 (5), pp. 1161–1165 (Russian).

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Notes:

In Russian. English translation: Sov. Math., Dokl. 30(1988), no. 8, pp. 666–668.

External Links:

ISSN 0002-3264, MathReview (Vojislav Marić), ZentralBlatt

Referenced by:

§32.11(iv).

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3: Gergő Nemes

… ►Nemes has research interests in asymptotic analysis, Écalle theory, exact WKB analysis, and special functions. ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …

4: 20 Theta Functions

Chapter 20 Theta Functions

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5: 28.16 Asymptotic Expansions for Large $q$

§28.16 Asymptotic Expansions for Large $q$

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28.16.1 $${\lambda }_{\nu }\left(h^2\right)\sim -2h^2+2sh-\frac{1}{8}(s^2+1)-\frac{1}{2^{7}h}(s^3+3s)-\frac{1}{2^{12}{h}^2}(5s^4+34s^2+9)-\frac{1}{2^{17}{h}^3}(33s^5+410s^3+405s)-\frac{1}{2^{20}{h}^4}(63s^6+1260s^4+2943s^2+486)-\frac{1}{2^{25}{h}^5}(527s^7+15617s^5+69001s^3+41607s)+\mathrm{\cdots }.$$

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Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $\sim$: Poincaré asymptotic expansion, $h$: parameter and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.16 and Ch.28

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6: Bibliography W

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The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

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External Links:

ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt

Referenced by:

§29.19(ii).

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Asymptotic behavior of stability regions for Hill’s equation. SIAM J. Appl. Math. 47 (5), pp. 941–958.

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External Links:

ISSN 0036-1399, Document, MathReview (Ekkehard Wagenführer), ZentralBlatt

Referenced by:

§28.29(iii).

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On the asymptotic behavior of the Fourier coefficients of Mathieu functions. J. Res. Nat. Inst. Standards Tech. 113 (1), pp. 11–15.

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External Links:

ISSN 1044-677X, FullText

Referenced by:

§28.4(vii).

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Exponential asymptotics of the Mittag-Leffler function. Constr. Approx. 18 (3), pp. 355–385.

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External Links:

ISSN 0176-4276, Document, MathReview, ZentralBlatt

Referenced by:

§10.46.

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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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7: Bibliography B

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Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

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External Links:

Document

Referenced by:

§33.22(iv).

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A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.

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Notes:

Double-precision Fortran, maximum accuracy 10S.

External Links:

Document, Code, ZentralBlatt

Referenced by:

1st item.

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Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

1st item, 4th item.

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Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.73(iv).

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Asymptotic behavior of the Pollaczek polynomials and their zeros. Stud. Appl. Math. 96, pp. 307–338.

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External Links:

ISSN 0022-2526, MathReview (Harold Exton), ZentralBlatt

Referenced by:

§18.35(iii).

Encodings:

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8: 10.75 Tables

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Achenbach (1986) tabulates $J_0\left(x\right)$, $J_1\left(x\right)$, $Y_0\left(x\right)$, $Y_1\left(x\right)$, $x=0(.1)8$, 20D or 18–20S.

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Olver (1960) tabulates $j_{n,m}$, $J_n^{\prime }\left(j_{n,m}\right)$, $j_{n,m}^{\prime }$, $J_n\left(j_{n,m}^{\prime }\right)$, $y_{n,m}$, $Y_n^{\prime }\left(y_{n,m}\right)$, $y_{n,m}^{\prime }$, $Y_n\left(y_{n,m}^{\prime }\right)$, $n=0(\frac{1}{2})20⁤\frac{1}{2}$, $m=1(1)50$, 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as $n\to \mathrm{\infty }$; see §10.21(viii), and more fully Olver (1954).

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Bickley et al. (1952) tabulates $x^{-n}{I}_n\left(x\right)$ or ${\mathrm{e}}^{-x}{I}_n\left(x\right)$, $x^n{K}_n\left(x\right)$ or ${\mathrm{e}}^x{K}_n\left(x\right)$, $n=2(1)20$, $x=0$(.01 or .1) 10(.1) 20, 8S; $I_n\left(x\right)$, $K_n\left(x\right)$, $n=0(1)20$, $x=0$ or $0.1(.1)20$, 10S.

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Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of $K_n\left(z\right)$ and $K_n^{\prime }\left(z\right)$, for $n=2(1)20$, 9S.

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Olver (1960) tabulates $a_{n,m}^{\prime }$, ${𝗃}_n\left(a_{n,m}^{\prime }\right)$, $b_{n,m}^{\prime }$, ${𝗒}_n\left(b_{n,m}^{\prime }\right)$, $n=1(1)20$, $m=1(1)50$, 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as $n\to \mathrm{\infty }$.

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9: Bibliography K

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Asymptotic behavior of the solutions of the Painlevé equation of the first kind. Differ. Uravn. 24 (10), pp. 1684–1695 (Russian).

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Notes:

In Russian. English translation: Differential Equations 24(1988), no. 10, pp. 1107–1115 (1989)

External Links:

ISSN 0374-0641, MathReview, ZentralBlatt

Referenced by:

§32.11(i).

Encodings:

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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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External Links:

ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt

Referenced by:

1st item.

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Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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External Links:

Document, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

§25.18(i).

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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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External Links:

ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt

Referenced by:

§32.11(iv).

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The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.

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External Links:

ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt

Referenced by:

§18.26(v), §18.26(v).

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10: 26.3 Lattice Paths: Binomial Coefficients

§26.3 Lattice Paths: Binomial Coefficients

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§26.3(i) Definitions

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§26.3(ii) Generating Functions

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§26.3(iii) Recurrence Relations

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§26.3(iv) Identities

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