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29 Lamé FunctionsLamé Functions

§29.3 Definitions and Basic Properties

Contents
  1. §29.3(i) Eigenvalues
  2. §29.3(ii) Distribution
  3. §29.3(iii) Continued Fractions
  4. §29.3(iv) Lamé Functions
  5. §29.3(v) Normalization
  6. §29.3(vi) Orthogonality
  7. §29.3(vii) Power Series

§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,,m1.

If ν is distinct from 0,1,,m1, then

29.3.6 (aνm(k2)bνm(k2))ν(ν1)(νm+1)>0.

If ν is a nonnegative integer, then

29.3.7 aνm(k2)+aννm(1k2)=ν(ν+1),
m=0,1,,ν,
29.3.8 bνm(k2)+bννm+1(1k2)=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 βpHαp1γpβp1Hαp2γp1βp2H=αpγp+1βp+1Hαp+1γp+2βp+2H,

where p is any nonnegative integer, and

29.3.11 αp={(ν1)(ν+2)k2,p=0,12(ν2p1)(ν+2p+2)k2,p1,
29.3.12 βp =4p2(2k2),
γp =12(ν2p+2)(ν+2p1)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 β0H. 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={2k2+12ν(ν+1)k2,p=0,(2p+1)2(2k2),p1,
29.3.14 αp =12(ν2p2)(ν+2p+3)k2,
γp =12(ν2p+1)(ν+2p)k2.

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

29.3.15 βp={2k212ν(ν+1)k2,p=0,(2p+1)2(2k2),p1,
29.3.16 αp =12(ν2p2)(ν+2p+3)k2,
γp =12(ν2p+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(ν2p3)(ν+2p+4)k2,
βp =(2p+2)2(2k2),
γp =12(ν2p)(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(zK)
period of
w(z)
dw/dz|z=0=dw/dz|z=K=0 aν2m(k2) 𝐸𝑐ν2m(z,k2) even even 2K
w(0)=dw/dz|z=K=0 aν2m+1(k2) 𝐸𝑐ν2m+1(z,k2) odd even 4K
dw/dz|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))2dx =14π,
0Kdn(x,k)(𝐸𝑐ν2m+1(x,k2))2dx =14π,
0Kdn(x,k)(𝐸𝑠ν2m+1(x,k2))2dx =14π,
0Kdn(x,k)(𝐸𝑠ν2m+2(x,k2))2dx =14π.

For dn(z,k) see §22.2.

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

§29.3(vi) Orthogonality

For mp,

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