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29 Lamé FunctionsNotation

§29.1 Special Notation

(For other notation see Notation for the Special Functions.)

m,n,p nonnegative integers.
x real variable.
z complex variable.
h,k,ν real parameters, 0<k<1, ν-12.
k 1-k2, 0<k<1.
K, K complete elliptic integrals of the first kind with moduli k,k, respectively (see §19.2(ii)).

All derivatives are denoted by differentials, not by primes.

The main functions treated in this chapter are the eigenvalues aν2m(k2), aν2m+1(k2), bν2m+1(k2), bν2m+2(k2), the Lamé functions Ecν2m(z,k2), Ecν2m+1(z,k2), Esν2m+1(z,k2), Esν2m+2(z,k2), and the Lamé polynomials uE2nm(z,k2), sE2n+1m(z,k2), cE2n+1m(z,k2), dE2n+1m(z,k2), scE2n+2m(z,k2), sdE2n+2m(z,k2), cdE2n+2m(z,k2), scdE2n+3m(z,k2). The notation for the eigenvalues and functions is due to Erdélyi et al. (1955, §15.5.1) and that for the polynomials is due to Arscott (1964b, §9.3.2). The normalization is that of Jansen (1977, §3.1).

Other notations that have been used are as follows: Ince (1940a) interchanges aν2m+1(k2) with bν2m+1(k2). The relation to the Lamé functions Lcν(m), Lsν(m)of Jansen (1977) is given by

Ecν2m(z,k2) =(-1)mLcν(2m)(ψ,k2),
Ecν2m+1(z,k2) =(-1)mLsν(2m+1)(ψ,k2),
Esν2m+1(z,k2) =(-1)mLcν(2m+1)(ψ,k2),
Esν2m+2(z,k2) =(-1)mLsν(2m+2)(ψ,k2),

where ψ=am(z,k); see §22.16(i). The relation to the Lamé functions Ecνm, Esνm of Ince (1940b) is given by

Ecν2m(z,k2) =cν2m(k2)Ecν2m(z,k2),
Ecν2m+1(z,k2) =cν2m+1(k2)Esν2m+1(z,k2),
Esν2m+1(z,k2) =sν2m+1(k2)Ecν2m+1(z,k2),
Esν2m+2(z,k2) =sν2m+2(k2)Esν2m+2(z,k2),

where the positive factors cνm(k2) and sνm(k2) are determined by

(cνm(k2))2 =4π0K(Ecνm(x,k2))2dx,
(sνm(k2))2 =4π0K(Esνm(x,k2))2dx.