Lamé functions

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21: 29.18 Mathematical Applications

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§29.18(i) Sphero-Conal Coordinates

… ►(29.18.5) is the differential equation of spherical Bessel functions (§10.47(i)), and (29.18.6), (29.18.7) agree with the Lamé equation (29.2.1). … ►

§29.18(iv) Other Applications

►Triebel (1965) gives applications of Lamé functions to the theory of conformal mappings. …

22: 29.8 Integral Equations

§29.8 Integral Equations

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29.8.2 $$\mu w(z_1)w(z_2)w(z_3)={\int }_{-2K}^{2K}{𝖯}_{\nu }\left(x\right)w(z)dz,$$

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $dx$: differential of $x$ , $\int$: integral, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $x$: real variable, $z$: complex variable, $k$: real parameter, $\nu$: real parameter, $w(z)$: solution and $\mu$

Referenced by:

§29.8

Permalink:

http://dlmf.nist.gov/29.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §29.8 and Ch.29

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29.8.7 $${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z_1,k^2)\frac{w_2(K)+w_2(-K)}{w_2(0)}=-k^2\mathrm{sn}(z_1,k){\int }_{-K}^K\mathrm{sn}(z,k)\frac{d{𝖯}_{\nu }\left(y\right)}{dy}{\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z,k^2)dz,$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\int$: integral, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $m$: nonnegative integer, $y$: real variable, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $w(z)$: solution

Permalink:

http://dlmf.nist.gov/29.8.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §29.8 and Ch.29

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29.8.8 $${\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(z_1,k^2)\frac{{d{w}_2(z)/dz|}_{z=K}+{d{w}_2(z)/dz|}_{z=-K}}{{d{w}_2(z)/dz|}_{z=0}}=\frac{k^2}{k^{\prime }}\mathrm{cn}(z_1,k){\int }_{-K}^K\mathrm{cn}(z,k)\frac{d{𝖯}_{\nu }\left(y\right)}{dy}{\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(z,k^2)dz,$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\int$: integral, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $m$: nonnegative integer, $y$: real variable, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $w(z)$: solution

Permalink:

http://dlmf.nist.gov/29.8.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §29.8 and Ch.29

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29.8.9 $${\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z_1,k^2)\frac{{d{w}_2(z)/dz|}_{z=K}-{d{w}_2(z)/dz|}_{z=-K}}{w_2(0)}=-\frac{k^4}{k^{\prime }}\mathrm{sn}(z_1,k)\mathrm{cn}(z_1,k){\int }_{-K}^K\mathrm{sn}(z,k)\mathrm{cn}(z,k)\frac{d^2{𝖯}_{\nu }\left(y\right)}{{dy}^2}{\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z,k^2)dz.$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\int$: integral, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $m$: nonnegative integer, $y$: real variable, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $w(z)$: solution

Permalink:

http://dlmf.nist.gov/29.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §29.8 and Ch.29

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23: 29.15 Fourier Series and Chebyshev Series

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29.15.18 $${\mathrm{𝑑𝐸}}_{2n+1}^m(z,k^2)=\mathrm{dn}(z,k)\left(\frac{1}{2}{C}_0+\sum _{p=1}^n{C}_{2p}\cos \left(2p\varphi \right)\right).$$

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Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝑑𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $\cos z$: cosine function, $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\varphi$: change of variable and $C_{2p+1}$: coefficents

Referenced by:

§29.15(i)

Permalink:

http://dlmf.nist.gov/29.15.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

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29.15.28 $${\mathrm{𝑠𝑑𝐸}}_{2n+2}^m(z,k^2)=\mathrm{dn}(z,k)\sum _{p=0}^n{C}_{2p+1}\cos \left((2p+1)\varphi \right).$$

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Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝑠𝑑𝐸}}_{2n+2}^m(z,k^2)$: Lamé polynomial, $\cos z$: cosine function, $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\varphi$: change of variable and $C_{2p+1}$: coefficents

Referenced by:

§29.15(i)

Permalink:

http://dlmf.nist.gov/29.15.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

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29.15.33 $${\mathrm{𝑐𝑑𝐸}}_{2n+2}^m(z,k^2)=\mathrm{dn}(z,k)\sum _{p=0}^n{D}_{2p+1}\sin \left((2p+1)\varphi \right).$$

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Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝑐𝑑𝐸}}_{2n+2}^m(z,k^2)$: Lamé polynomial, $\sin z$: sine function, $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\varphi$: change of variable and $D_{2p}$: coefficents

Referenced by:

§29.15(i)

Permalink:

http://dlmf.nist.gov/29.15.E33

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

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29.15.38 $${\mathrm{𝑠𝑐𝑑𝐸}}_{2n+3}^m(z,k^2)=\mathrm{dn}(z,k)\sum _{p=0}^n{D}_{2p+2}\sin \left((2p+2)\varphi \right).$$

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Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝑠𝑐𝑑𝐸}}_{2n+3}^m(z,k^2)$: Lamé polynomial, $\sin z$: sine function, $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\varphi$: change of variable and $D_{2p}$: coefficents

Referenced by:

§29.15(i)

Permalink:

http://dlmf.nist.gov/29.15.E38

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

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29.15.49 $${\mathrm{𝑐𝑑𝐸}}_{2n+2}^m(z,k^2)=\mathrm{cn}(z,k)\mathrm{dn}(z,k)\sum _{p=0}^n{D}_{2p+1}{U}_{2p}\left(\mathrm{sn}(z,k)\right),$$

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Symbols:

$U_n\left(x\right)$: Chebyshev polynomial of the second kind, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝑐𝑑𝐸}}_{2n+2}^m(z,k^2)$: Lamé polynomial, $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter and $D_{2p}$: coefficents

Permalink:

http://dlmf.nist.gov/29.15.E49

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(ii), §29.15 and Ch.29

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24: Bibliography I

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The periodic Lamé functions. Proc. Roy. Soc. Edinburgh 60, pp. 47–63.

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External Links:

MathReview (H. Bateman), ZentralBlatt

Referenced by:

§29.1, §29.15(ii), 1st item, §29.7(i).

Encodings:

BibTeX

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Further investigations into the periodic Lamé functions. Proc. Roy. Soc. Edinburgh 60, pp. 83–99.

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External Links:

MathReview (H. Bateman), ZentralBlatt

Referenced by:

§29.1, §29.17(ii), §29.3(iii), §29.6(i), §29.6(ii), §29.6(iii), §29.6(iv), Ch.29, Notations E, Notations E, Notations E, Notations E.

Encodings:

BibTeX

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25: 31.8 Solutions via Quadratures

… ►For $𝐦=(m_0,0,0,0)$, these solutions reduce to Hermite’s solutions (Whittaker and Watson (1927, §23.7)) of the Lamé equation in its algebraic form. …

26: Bibliography E

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On Lamé functions. Philos. Mag. (7) 31, pp. 123–130.

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External Links:

MathReview (G. Szegő), ZentralBlatt

Referenced by:

§29.3(ii).

Encodings:

BibTeX

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On algebraic Lamé functions. Philos. Mag. (7) 32, pp. 348–350.

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External Links:

MathReview (G. Szegő), ZentralBlatt

Referenced by:

§29.17(ii).

Encodings:

BibTeX

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27: Bibliography V

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Integral relations for Lamé functions. SIAM J. Math. Anal. 13 (6), pp. 978–987.

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External Links:

ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.8, §29.8.

Encodings:

BibTeX

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Integral representations for products of Lamé functions by use of fundamental solutions. SIAM J. Math. Anal. 15 (3), pp. 559–569.

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External Links:

ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.8.

Encodings:

BibTeX

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28: 28.34 Methods of Computation

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Asymptotic approximations by zeros of orthogonal polynomials of increasing degree. See Volkmer (2008). This method also applies to eigenvalues of the Whittaker–Hill equation (§28.31(i)) and eigenvalues of Lamé functions (§29.3(i)).

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29: Bibliography R

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On the computation of Lamé functions, of eigenvalues and eigenfunctions of some potential operators. Z. Angew. Math. Mech. 78 (1), pp. 66–72.

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External Links:

ISSN 0044-2267, Document, MathReview Entry, ZentralBlatt

Referenced by:

§29.20(ii).

Encodings:

BibTeX

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Some Applications of the Lamé Function Solutions of the Linearised Supersonic Flow Equations. Technical Reports and Memoranda Technical Report 2865, Aeronautical Research Council (Great Britain).

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Notes:

Reprinted by Her Majesty’s Stationery Office, London, 1962.

External Links:

MathReview (P. Germain)

Referenced by:

§29.19(ii).

Encodings:

BibTeX

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30: Bibliography J

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Simple-periodic and Non-periodic Lamé Functions. Mathematical Centre Tracts, No. 72, Mathematisch Centrum, Amsterdam.

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External Links:

ISBN 90-6196-130-0, MathReview (V. P. Madan)

Referenced by:

§29.1, §29.1, §29.18(i), §29.19(i), §29.20(i), §29.20(i), §29.22(ii), §29.3(v), §29.6(i), §29.6(ii), §29.6(iii), §29.6(iv), Ch.29, Notations L, Notations L, Notations L, Notations L.

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BibTeX

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