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29 Lamé FunctionsLamé Functions

§29.3 Definitions and Basic Properties

Contents

§29.3(i) Eigenvalues

For each pair of values of ν and k there are four infinite unbounded sets of real eigenvalues h for which equation (29.2.1) has even or odd solutions with periods 2K or 4K. They are denoted by aν2m(k2), aν2m+1(k2), bν2m+1(k2), bν2m+2(k2), where m=0,1,2,; see Table 29.3.1.

Table 29.3.1: Eigenvalues of Lamé’s equation.
eigenvalue h parity period
aν2m(k2) even 2K
aν2m+1(k2) odd 4K
bν2m+1(k2) even 4K
bν2m+2(k2) odd 2K

§29.3(ii) Distribution

The eigenvalues interlace according to

29.3.1 aνm(k2) <aνm+1(k2),
29.3.2 aνm(k2) <bνm+1(k2),
29.3.3 bνm(k2) <bνm+1(k2),
29.3.4 bνm(k2) <aνm+1(k2).

The eigenvalues coalesce according to

29.3.5 aνm(k2)=bνm(k2),
ν=0,1,,m-1.

If ν is distinct from 0,1,,m-1, then

29.3.6 (aνm(k2)-bνm(k2))ν(ν-1)(ν-m+1)>0.

If ν is a nonnegative integer, then

29.3.7 aνm(k2)+aνν-m(1-k2)=ν(ν+1),
m=0,1,,ν,
29.3.8 bνm(k2)+bνν-m+1(1-k2)=ν(ν+1),
m=1,2,,ν.

For the special case k=k=1/2 see Erdélyi et al. (1955, §15.5.2).

§29.3(iii) Continued Fractions

The quantity

29.3.9 H=2aν2m(k2)-ν(ν+1)k2

satisfies the continued-fraction equation

29.3.10 βp-H-αp-1γpβp-1-H-αp-2γp-1βp-2-H-=αpγp+1βp+1-H-αp+1γp+2βp+2-H-,

where p is any nonnegative integer, and

29.3.11 αp={(ν-1)(ν+2)k2,p=0,12(ν-2p-1)(ν+2p+2)k2,p1,
29.3.12 βp =4p2(2-k2),
γp =12(ν-2p+2)(ν+2p-1)k2.

The continued fraction following the second negative sign on the left-hand side of (29.3.10) is finite: it equals 0 if p=0, and if p>0, then the last denominator is β0-H. If ν is a nonnegative integer and 2pν, then the continued fraction on the right-hand side of (29.3.10) terminates, and (29.3.10) has only the solutions (29.3.9) with 2mν. If ν is a nonnegative integer and 2p>ν, then (29.3.10) has only the solutions (29.3.9) with 2m>ν.

The quantity H=2aν2m+1(k2)-ν(ν+1)k2 satisfies equation (29.3.10) with

29.3.13 βp={2-k2+12ν(ν+1)k2,p=0,(2p+1)2(2-k2),p1,
29.3.14 αp =12(ν-2p-2)(ν+2p+3)k2,
γp =12(ν-2p+1)(ν+2p)k2.

The quantity H=2bν2m+1(k2)-ν(ν+1)k2 satisfies equation (29.3.10) with

29.3.15 βp={2-k2-12ν(ν+1)k2,p=0,(2p+1)2(2-k2),p1,
29.3.16 αp =12(ν-2p-2)(ν+2p+3)k2,
γp =12(ν-2p+1)(ν+2p)k2.

The quantity H=2bν2m+2(k2)-ν(ν+1)k2 satisfies equation (29.3.10) with

29.3.17 αp =12(ν-2p-3)(ν+2p+4)k2,
βp =(2p+2)2(2-k2),
γp =12(ν-2p)(ν+2p+1)k2.

§29.3(iv) Lamé Functions

The eigenfunctions corresponding to the eigenvalues of §29.3(i) are denoted by Ecν2m(z,k2), Ecν2m+1(z,k2), Esν2m+1(z,k2), Esν2m+2(z,k2). They are called Lamé functions with real periods and of order ν, or more simply, Lamé functions. See Table 29.3.2. In this table the nonnegative integer m corresponds to the number of zeros of each Lamé function in (0,K), whereas the superscripts 2m, 2m+1, or 2m+2 correspond to the number of zeros in [0,2K).

Table 29.3.2: Lamé functions.
boundary conditions
eigenvalue
h
eigenfunction
w(z)
parity of
w(z)
parity of
w(z-K)
period of
w(z)
w/z|z=0=w/z|z=K=0 aν2m(k2) Ecν2m(z,k2) even even 2K
w(0)=w/z|z=K=0 aν2m+1(k2) Ecν2m+1(z,k2) odd even 4K
w/z|z=0=w(K)=0 bν2m+1(k2) Esν2m+1(z,k2) even odd 4K
w(0)=w(K)=0 bν2m+2(k2) Esν2m+2(z,k2) odd odd 2K

§29.3(v) Normalization

29.3.18 0Kdn(x,k)(Ecν2m(x,k2))2x =14π,
0Kdn(x,k)(Ecν2m+1(x,k2))2x =14π,
0Kdn(x,k)(Esν2m+1(x,k2))2x =14π,
0Kdn(x,k)(Esν2m+2(x,k2))2x =14π.

For dn(z,k) see §22.2.

To complete the definitions, Ecνm(K,k2) is positive and Esνm(z,k2)/z|z=K is negative.

§29.3(vi) Orthogonality

For mp,

29.3.19 0KEcν2m(x,k2)Ecν2p(x,k2)x =0,
0KEcν2m+1(x,k2)Ecν2p+1(x,k2)x =0,
0KEsν2m+1(x,k2)Esν2p+1(x,k2)x =0,
0KEsν2m+2(x,k2)Esν2p+2(x,k2)x =0.

For the values of these integrals when m=p see §29.6.

§29.3(vii) Power Series

For power-series expansions of the eigenvalues see Volkmer (2004b).