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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, ν12.

k

1k2, 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.