Lamé functions

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41: 29.11 Lamé Wave Equation

§29.11 Lamé Wave Equation

►The Lamé (or ellipsoidal) wave equation is given by ►

29.11.1 $$\frac{d^{2}w}{{dz}^2}+(h-\nu (\nu +1)k^2{\mathrm{sn}}^2(z,k)+k^2{\omega }^2{\mathrm{sn}}^4(z,k))w=0,$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $h$: real parameter, $k$: real parameter, $\nu$: real parameter, $\omega$: parameter and $w(z)$: solution

Referenced by:

§29.11, §29.11, §29.18(ii), §29.7(ii)

Permalink:

http://dlmf.nist.gov/29.11.E1

Encodings:

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See also:

Annotations for §29.11 and Ch.29

… ►In the case $\omega =0$, (29.11.1) reduces to Lamé’s equation (29.2.1). …

42: 29.14 Orthogonality

§29.14 Orthogonality

►Lamé polynomials are orthogonal in two ways. …Secondly, the system of functions …where … ►

29.14.4 $${\mathrm{𝑠𝐸}}_{2n+1}^m(s,k^2){\mathrm{𝑠𝐸}}_{2n+1}^m(K+\mathrm{i}t,k^2),$$

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Symbols:

${\mathrm{𝑠𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter

Referenced by:

§29.14

Permalink:

http://dlmf.nist.gov/29.14.E4

Encodings:

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See also:

Annotations for §29.14 and Ch.29

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43: 29.16 Asymptotic Expansions

§29.16 Asymptotic Expansions

►Hargrave and Sleeman (1977) give asymptotic approximations for Lamé polynomials and their eigenvalues, including error bounds. The approximations for Lamé polynomials hold uniformly on the rectangle $0\le \mathrm{\Re }z\le K$, $0\le \mathrm{\Im }z\le K^{\prime }$, when $nk$ and $n{k}^{\prime }$ assume large real values. The approximating functions are exponential, trigonometric, and parabolic cylinder functions.

44: Bibliography F

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Generalized parabolic cylinder functions. Asymptotic Anal. 5 (6), pp. 517–531.

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External Links:

ISSN 0921-7134, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§12.15.

Encodings:

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The Lamé wave equation. Uspekhi Mat. Nauk 44 (1(265)), pp. 123–144, 248 (Russian).

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Notes:

In Russian. English translation: Russian Math. Surveys, 44(1989), no. 1, pp. 153–180

External Links:

ISSN 0042-1316, Document, MathReview (R. G. Langebartel), ZentralBlatt

Referenced by:

§29.11.

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Complex zeros of the error function and of the complementary error function. Math. Comp. 27 (122), pp. 401–407.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§7.13(i), §7.13(ii), 1st item.

Encodings:

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Expansions of hypergeometric functions in hypergeometric functions. Math. Comp. 15 (76), pp. 390–395.

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External Links:

FullText, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.10.

Encodings:

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Spheroidal Wave Functions. Stanford University Press, Stanford, CA.

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External Links:

MathReview (J. C. P. Miller), ZentralBlatt

Referenced by:

§30.1, §30.10, 2nd item, Ch.30.

Encodings:

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45: Bibliography D

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Tables of Higher Mathematical Functions I. Principia Press, Bloomington, Indiana.

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External Links:

ZentralBlatt

Referenced by:

§5.1, Notations P.

Encodings:

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Integral representations for elliptic functions. J. Math. Anal. Appl. 316 (1), pp. 142–160.

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External Links:

ISSN 0022-247X, Document, MathReview Entry, ZentralBlatt

Referenced by:

§23.11.

Encodings:

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Bernoulli Numbers and Confluent Hypergeometric Functions. In Number Theory for the Millennium, I (Urbana, IL, 2000), pp. 343–363.

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External Links:

ISBN 1-56881-126-8, MathReview (Arnold M. Adelberg), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

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Lamé instantons. J. High Energy Phys. 2000 (1), pp. Paper 19, 8.

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Notes:

(electronic only, article id. JHEP01(2000)019, 8 pp.).

External Links:

ISSN 1029-8479, Document, MathReview, ZentralBlatt

Referenced by:

§22.11.

Encodings:

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Functions with given moments and weight functions for orthogonal polynomials. Rocky Mountain J. Math. 23, pp. 87–104.

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External Links:

ISSN 0035-7596, MathReview (Olav Njåstad), ZentralBlatt

Referenced by:

§18.34(ii).

Encodings:

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46: Bibliography P

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Tables of Digamma and Trigamma Functions. In Tracts for Computers, No. 1, K. Pearson (Ed.),

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External Links:

ZentralBlatt

Referenced by:

§5.1, Notations F.

Encodings:

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Exponential asymptotics of the Mittag-Leffler function. Proc. Roy. Soc. London Ser. A 458, pp. 3041–3052.

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External Links:

ISSN 1364-5021, Document, MathReview (Roderick Wong), ZentralBlatt

Referenced by:

§10.46.

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A new basis for the representation of the rotation group. Lamé and Heun polynomials. J. Mathematical Phys. 14 (8), pp. 1130–1139.

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External Links:

Document, ZentralBlatt

Referenced by:

§29.18(iv).

Encodings:

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Tables of the Incomplete Beta-function. 2nd edition, Published for the Biometrika Trustees at the Cambridge University Press, Cambridge.

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External Links:

MathReview, ZentralBlatt

Referenced by:

1st item.

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Bounds for ratios of modified Bessel functions. Integral Transform. Spec. Funct. 9 (4), pp. 293–298.

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External Links:

ISSN 1065-2469, Document, MathReview, ZentralBlatt

Referenced by:

§10.37.

Encodings:

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47: Bibliography H

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Certain integrals that contain a probability function and degenerate hypergeometric functions. Bul. Akad. S̆tiince RSS Moldoven 1969 (2), pp. 40–47 (Russian).

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External Links:

MathReview (Wazir Hasan Abdi)

Referenced by:

§7.14(iii).

Encodings:

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Some integrals that contain a probability function and hypergeometric functions. Bul. Akad. Štiince RSS Moldoven 1970 (1), pp. 49–62 (Russian).

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External Links:

MathReview (R. G. Langebartel)

Referenced by:

§7.14(iii).

Encodings:

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Expansions for the probability function in series of Čebyšev polynomials and Bessel functions. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).

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External Links:

MathReview (V. L. Makarov)

Referenced by:

§7.14(iii).

Encodings:

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Sums with cylindrical functions that reduce to the probability function and to related functions. Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).

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External Links:

MathReview (M. Lawrence Glasser)

Referenced by:

§7.14(iii).

Encodings:

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Lamé polynomials of large order. SIAM J. Math. Anal. 8 (5), pp. 800–842.

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External Links:

Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.16.

Encodings:

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