29 Lamé Functions Lamé Polynomials 29.11 Lamé Wave Equation 29.13 Graphics

§29.12 Definitions

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Keywords:: Lamé polynomials, definition
Referenced by:: §29.12(i)
Permalink:: http://dlmf.nist.gov/29.12
See also:: Annotations for Ch.29

§29.12(i) Elliptic-Function Form
§29.12(ii) Algebraic Form
§29.12(iii) Zeros

§29.12(i) Elliptic-Function Form

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Keywords:: Lamé functions, Lamé polynomials, elliptic-function form, notation, periodicity, relation to Lamé functions, relations to Lamé polynomials, zeros
Notes:: See Arscott (1964b, Chapter 9).
Referenced by:: §29.14, §29.6(i)
Permalink:: http://dlmf.nist.gov/29.12.i
See also:: Annotations for §29.12 and Ch.29

Throughout §§29.12–29.16 the order $\nu$ in the differential equation (29.2.1) is assumed to be a nonnegative integer.

The Lamé functions $\mathit{Ec}^{m}_{\nu}\left(z,k^{2}\right)$ , $m=0,1,\dots,\nu$ , and $\mathit{Es}^{m}_{\nu}\left(z,k^{2}\right)$ , $m=1,2,\dots,\nu$ , are called the Lamé polynomials. There are eight types of Lamé polynomials, defined as follows:

29.12.1 $\mathit{uE}^{m}_{2n}\left(z,k^{2}\right)=\mathit{Ec}^{2m}_{2n}\left(z,k^{2}% \right),$

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Defines:: $\mathit{uE}^{\NVar{m}}_{2\NVar{n}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial
Symbols:: $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, : nonnegative integer, : nonnegative integer, : complex variable and : real parameter
Referenced by:: §29.12(i)
Permalink:: http://dlmf.nist.gov/29.12.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §29.12(i), §29.12 and Ch.29

29.12.2 $\mathit{sE}^{m}_{2n+1}\left(z,k^{2}\right)=\mathit{Ec}^{2m+1}_{2n+1}\left(z,k^% {2}\right),$

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Defines:: $\mathit{sE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial
Symbols:: $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, : nonnegative integer, : nonnegative integer, : complex variable and : real parameter
Permalink:: http://dlmf.nist.gov/29.12.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §29.12(i), §29.12 and Ch.29

29.12.3 $\mathit{cE}^{m}_{2n+1}\left(z,k^{2}\right)=\mathit{Es}^{2m+1}_{2n+1}\left(z,k^% {2}\right),$

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Defines:: $\mathit{cE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial
Symbols:: $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, : nonnegative integer, : nonnegative integer, : complex variable and : real parameter
Permalink:: http://dlmf.nist.gov/29.12.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §29.12(i), §29.12 and Ch.29

29.12.4 $\mathit{dE}^{m}_{2n+1}\left(z,k^{2}\right)=\mathit{Ec}^{2m}_{2n+1}\left(z,k^{2% }\right),$

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Defines:: $\mathit{dE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial
Symbols:: $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, : nonnegative integer, : nonnegative integer, : complex variable and : real parameter
Permalink:: http://dlmf.nist.gov/29.12.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §29.12(i), §29.12 and Ch.29

29.12.5 $\mathit{scE}^{m}_{2n+2}\left(z,k^{2}\right)=\mathit{Es}^{2m+2}_{2n+2}\left(z,k% ^{2}\right),$

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Defines:: $\mathit{scE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial
Symbols:: $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, : nonnegative integer, : nonnegative integer, : complex variable and : real parameter
Permalink:: http://dlmf.nist.gov/29.12.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §29.12(i), §29.12 and Ch.29

29.12.6 $\mathit{sdE}^{m}_{2n+2}\left(z,k^{2}\right)=\mathit{Ec}^{2m+1}_{2n+2}\left(z,k% ^{2}\right),$

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Defines:: $\mathit{sdE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial
Symbols:: $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, : nonnegative integer, : nonnegative integer, : complex variable and : real parameter
Permalink:: http://dlmf.nist.gov/29.12.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §29.12(i), §29.12 and Ch.29

29.12.7 $\mathit{cdE}^{m}_{2n+2}\left(z,k^{2}\right)=\mathit{Es}^{2m+1}_{2n+2}\left(z,k% ^{2}\right),$

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Defines:: $\mathit{cdE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial
Symbols:: $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, : nonnegative integer, : nonnegative integer, : complex variable and : real parameter
Permalink:: http://dlmf.nist.gov/29.12.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §29.12(i), §29.12 and Ch.29

29.12.8 $\mathit{scdE}^{m}_{2n+3}\left(z,k^{2}\right)=\mathit{Es}^{2m+2}_{2n+3}\left(z,% k^{2}\right),$

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Defines:: $\mathit{scdE}^{\NVar{m}}_{2\NVar{n}+3}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial
Symbols:: $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, : nonnegative integer, : nonnegative integer, : complex variable and : real parameter
Referenced by:: §29.12(i)
Permalink:: http://dlmf.nist.gov/29.12.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §29.12(i), §29.12 and Ch.29

where $n=0,1,2,\dots$ , $m=0,1,2,\dots,n$ . These functions are polynomials in $\operatorname{sn}\left(z,k\right)$ , $\operatorname{cn}\left(z,k\right)$ , and $\operatorname{dn}\left(z,k\right)$ . In consequence they are doubly-periodic meromorphic functions of .

The superscript on the left-hand sides of (29.12.1)–(29.12.8) agrees with the number of -zeros of each Lamé polynomial in the interval , while is the number of -zeros in the open line segment from to $K+\mathrm{i}{K^{\prime}}$ .

The prefixes $\mathit{u}$ , $\mathit{s}$ , $\mathit{c}$ , $\mathit{d}$ , $\mathit{sc}$ , $\mathit{sd}$ , $\mathit{cd}$ , $\mathit{scd}$ indicate the type of the polynomial form of the Lamé polynomial; compare the 3rd and 4th columns in Table 29.12.1. In the fourth column the variable and modulus of the Jacobian elliptic functions have been suppressed, and $P({\operatorname{sn}}^{2})$ denotes a polynomial of degree in ${\operatorname{sn}}^{2}\left(z,k\right)$ (different for each type). For the determination of the coefficients of the ’s see §29.15(ii).

Table 29.12.1: Lamé polynomials.

$\nu$

eigenvalue

eigenfunction

polynomial

form

real

period

imag.

period

parity of

$w(z-K-\mathrm{i}{K^{\prime}})$

$a^{2m}_{\nu}\left(k^{2}\right)$

$\mathit{uE}^{m}_{\nu}\left(z,k^{2}\right)$

$P({\operatorname{sn}}^{2})$

$2\mathrm{i}{K^{\prime}}$

even

$2n+1\;$

$a^{2m+1}_{\nu}\left(k^{2}\right)$

$\mathit{sE}^{m}_{\nu}\left(z,k^{2}\right)$

$\operatorname{sn}P({\operatorname{sn}}^{2})$

$2\mathrm{i}{K^{\prime}}$

odd

even

$2n+1\;$

$b^{2m+1}_{\nu}\left(k^{2}\right)$

$\mathit{cE}^{m}_{\nu}\left(z,k^{2}\right)$

$\operatorname{cn}P({\operatorname{sn}}^{2})$

$4\mathrm{i}{K^{\prime}}$

even

odd

even

$2n+1\;$

$a^{2m}_{\nu}\left(k^{2}\right)$

$\mathit{dE}^{m}_{\nu}\left(z,k^{2}\right)$

$\operatorname{dn}P({\operatorname{sn}}^{2})$

$4\mathrm{i}{K^{\prime}}$

even

odd

$2n+2\;$

$b^{2m+2}_{\nu}\left(k^{2}\right)$

$\mathit{scE}^{m}_{\nu}\left(z,k^{2}\right)$

$\operatorname{sn}\operatorname{cn}P({\operatorname{sn}}^{2})$

$4\mathrm{i}{K^{\prime}}$

odd

even

$2n+2\;$

$a^{2m+1}_{\nu}\left(k^{2}\right)$

$\mathit{sdE}^{m}_{\nu}\left(z,k^{2}\right)$

$\operatorname{sn}\operatorname{dn}P({\operatorname{sn}}^{2})$

$4\mathrm{i}{K^{\prime}}$

odd

even

odd

$2n+2\;$

$b^{2m+1}_{\nu}\left(k^{2}\right)$

$\mathit{cdE}^{m}_{\nu}\left(z,k^{2}\right)$

$\operatorname{cn}\operatorname{dn}P({\operatorname{sn}}^{2})$

$2\mathrm{i}{K^{\prime}}$

even

odd

$2n+3\;$

$b^{2m+2}_{\nu}\left(k^{2}\right)$

$\mathit{scdE}^{m}_{\nu}\left(z,k^{2}\right)$

$\operatorname{sn}\operatorname{cn}\operatorname{dn}P({\operatorname{sn}}^{2})$

$2\mathrm{i}{K^{\prime}}$

odd

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Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $a^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, $b^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, $\mathit{cE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, $\mathit{cdE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, $\mathit{dE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, $\mathit{sE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, $\mathit{scE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, $\mathit{scdE}^{\NVar{m}}_{2\NVar{n}+3}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, $\mathit{sdE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, $\mathit{uE}^{\NVar{m}}_{2\NVar{n}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$ : imaginary unit, : nonnegative integer, : nonnegative integer, : complex variable, : real parameter, : real parameter, $\nu$ : real parameter, : polynomial and : eigenfunction
Keywords:: Lamé polynomials, notation
Referenced by:: §29.12(i), §29.12(ii), §29.15(ii), 2nd item
Permalink:: http://dlmf.nist.gov/29.12.T1
See also:: Annotations for §29.12(i), §29.12 and Ch.29

§29.12(ii) Algebraic Form

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Keywords:: Lamé polynomials, algebraic form
Notes:: See Arscott (1964b, §9.3.2).
Permalink:: http://dlmf.nist.gov/29.12.ii
See also:: Annotations for §29.12 and Ch.29

With the substitution $\xi={\operatorname{sn}}^{2}\left(z,k\right)$ every Lamé polynomial in Table 29.12.1 can be written in the form

29.12.9 $\xi^{\rho}(\xi-1)^{\sigma}(\xi-k^{-2})^{\tau}P(\xi),$

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Symbols:: : real parameter, $\xi={\operatorname{sn}}^{2}\left(z,k\right)$ : substitution, (0 or $\tfrac{1}{2}$ ): parameter $\rho$ , (0 or $\tfrac{1}{2}$ ): parameter $\sigma$ , (0 or $\tfrac{1}{2}$ ): parameter $\tau$ and $P(\xi)$ : polynomial
Referenced by:: §29.12(ii), §29.12(iii)
Permalink:: http://dlmf.nist.gov/29.12.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §29.12(ii), §29.12 and Ch.29

where $\rho$ , $\sigma$ , $\tau$ are either 0 or $\frac{1}{2}$ . The polynomial $P(\xi)$ is of degree and has zeros (all simple) in and zeros (all simple) in $(1,k^{-2})$ . The functions (29.12.9) satisfy (29.2.2).

§29.12(iii) Zeros

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Keywords:: Lamé polynomials, applications, electrostatic interpretation, physical, zeros
Notes:: See Whittaker and Watson (1927, §23.46) and Arscott (1964b, §9.2.1).
Referenced by:: §29.19(ii)
Permalink:: http://dlmf.nist.gov/29.12.iii
See also:: Annotations for §29.12 and Ch.29

Let $\xi_{1},\xi_{2},\dots,\xi_{n}$ denote the zeros of the polynomial in (29.12.9) arranged according to

29.12.10 $0<\xi_{1}<\cdots<\xi_{m}<1<\xi_{m+1}<\cdots<\xi_{n}<k^{-2}.$

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Symbols:: : nonnegative integer, : nonnegative integer, : real parameter and $\xi_{n}$ : zeros
Permalink:: http://dlmf.nist.gov/29.12.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §29.12(iii), §29.12 and Ch.29

Then the function

29.12.11 $g(t_{1},t_{2},\dots,t_{n})=\left(\prod_{p=1}^{n}t_{p}^{\rho+\frac{1}{4}}|t_{p}% -1|^{\sigma+\frac{1}{4}}(k^{-2}-t_{p})^{\tau+\frac{1}{4}}\right)\prod_{q<r}(t_% {r}-t_{q}),$

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Defines:: $g(t_{1},\ldots,t_{n})$ : function (locally)
Symbols:: : nonnegative integer, : nonnegative integer, : real parameter, (0 or $\tfrac{1}{2}$ ): parameter $\rho$ , (0 or $\tfrac{1}{2}$ ): parameter $\sigma$ and (0 or $\tfrac{1}{2}$ ): parameter $\tau$
Referenced by:: §29.20(iii)
Permalink:: http://dlmf.nist.gov/29.12.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §29.12(iii), §29.12 and Ch.29

defined for $(t_{1},t_{2},\dots,t_{n})$ with

29.12.12 $0\leq t_{1}\leq\cdots\leq t_{m}\leq 1\leq t_{m+1}\leq\cdots\leq t_{n}\leq k^{-% 2},$

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Symbols:: : nonnegative integer, : nonnegative integer and : real parameter
Referenced by:: §29.12(iii)
Permalink:: http://dlmf.nist.gov/29.12.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §29.12(iii), §29.12 and Ch.29

attains its absolute maximum iff $t_{j}=\xi_{j}$ , $j=1,2,\dots,n$ . Moreover,

29.12.13 ${\frac{\rho+\frac{1}{4}}{\xi_{p}}+\frac{\sigma+\frac{1}{4}}{\xi_{p}-1}+\frac{% \tau+\frac{1}{4}}{\xi_{p}-k^{-2}}+\sum_{\begin{subarray}{c}q=1\\ q\neq p\end{subarray}}^{n}\frac{1}{\xi_{p}-\xi_{q}}=0},$ $p=1,2,\dots,n$ .

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Symbols:: : nonnegative integer, : nonnegative integer, : real parameter, (0 or $\tfrac{1}{2}$ ): parameter $\rho$ , (0 or $\tfrac{1}{2}$ ): parameter $\sigma$ , (0 or $\tfrac{1}{2}$ ): parameter $\tau$ and $\xi_{n}$ : zeros
Referenced by:: §29.20(iii)
Permalink:: http://dlmf.nist.gov/29.12.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §29.12(iii), §29.12 and Ch.29

This result admits the following electrostatic interpretation: Given three point masses fixed at , , and $t=k^{-2}$ with positive charges $\rho+\tfrac{1}{4}$ , $\sigma+\tfrac{1}{4}$ , and $\tau+\tfrac{1}{4}$ , respectively, and movable point masses at $t_{1},t_{2},\dots,t_{n}$ arranged according to (29.12.12) with unit positive charges, the equilibrium position is attained when $t_{j}=\xi_{j}$ for $j=1,2,\dots,n$ .

29.11 Lamé Wave Equation 29.13 Graphics

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§29.12 Definitions

Contents

§29.12(i) Elliptic-Function Form

§29.12(ii) Algebraic Form

§29.12(iii) Zeros