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29 Lamé FunctionsLamé Polynomials

§29.15 Fourier Series and Chebyshev Series

Contents

§29.15(i) Fourier Coefficients

Polynomial uE2nm(z,k2)

When ν=2n, m=0,1,,n, the Fourier series (29.6.1) terminates:

29.15.1 uE2nm(z,k2)=12A0+p=1nA2pcos(2pϕ).

A convenient way of constructing the coefficients, together with the eigenvalues, is as follows. Equations (29.6.4), with p=1,2,,n, (29.6.3), and A2n+2=0 can be cast as an algebraic eigenvalue problem in the following way. Let

29.15.2 M=[β0α000γ1β1α100γn-1βn-1αn-100γnβn]

be the tridiagonal matrix with αp, βp, γp as in (29.3.11), (29.3.12). Let the eigenvalues of M be Hp with

29.15.3 H0<H1<<Hn,

and also let

29.15.4 [A0,A2,,A2n]T

be the eigenvector corresponding to Hm and normalized so that

29.15.5 12A02+p=1nA2p2=1

and

29.15.6 12A0+p=1nA2p>0.

Then

29.15.7 aν2m(k2)=12(Hm+ν(ν+1)k2),

and (29.15.1) applies, with ϕ again defined as in (29.2.5).

Polynomial sE2n+1m(z,k2)

When ν=2n+1, m=0,1,,n, the Fourier series (29.6.16) terminates:

29.15.8 sE2n+1m(z,k2)=p=0nA2p+1cos((2p+1)ϕ).

In (29.15.2) replace αp, βp, and γp as in (29.3.13), (29.3.14). Also, replace (29.15.4), (29.15.5), (29.15.6) by

29.15.9 [A1,A3,,A2n+1]T,
29.15.10 p=0nA2p+12=1,
29.15.11 p=0nA2p+1>0.

Then

29.15.12 aν2m+1(k2)=12(Hm+ν(ν+1)k2),

and (29.15.8) applies.

Polynomial cE2n+1m(z,k2)

When ν=2n+1, m=0,1,,n, the Fourier series (29.6.31) terminates:

29.15.13 cE2n+1m(z,k2)=p=0nB2p+1sin((2p+1)ϕ).

In (29.15.2) replace αp, βp, and γp as in (29.3.15), (29.3.16). Also, replace (29.15.4), (29.15.5), (29.15.6) by

29.15.14 [B1,B3,,B2n+1]T,
29.15.15 p=0nB2p+12=1,
29.15.16 p=0n(2p+1)B2p+1>0.

Then

29.15.17 bν2m+1(k2)=12(Hm+ν(ν+1)k2),

and (29.15.13) applies.

Polynomial dE2n+1m(z,k2)

When ν=2n+1, m=0,1,,n, the Fourier series (29.6.8) terminates:

29.15.18 dE2n+1m(z,k2)=dn(z,k)(12C0+p=1nC2pcos(2pϕ)).

In (29.15.2) replace αp, βp, and γp as in (29.6.11). Also, replace (29.15.4), (29.15.5), (29.15.6) by

29.15.19 [C0,C2,,C2n]T,
29.15.20 (1-12k2)(12C02+p=1nC2p2)-12k2p=0n-1C2pC2p+2=1,
29.15.21 12C0+p=1nC2p>0.

Then

29.15.22 aν2m(k2)=12(Hm+ν(ν+1)k2),

and (29.15.18) applies.

Polynomial scE2n+2m(z,k2)

When ν=2n+2, m=0,1,,n, the Fourier series (29.6.46) terminates:

29.15.23 scE2n+2m(z,k2)=p=0nB2p+2sin((2p+2)ϕ).

In (29.15.2) replace αp, βp, and γp as in (29.3.17). Also replace (29.15.4), (29.15.5), (29.15.6) by

29.15.24 [B2,B4,,B2n+2]T,
29.15.25 p=0nB2p+22=1,
29.15.26 p=0n(2p+2)B2p+2>0.

Then

29.15.27 bν2m+2(k2)=12(Hm+ν(ν+1)k2),

and (29.15.23) applies.

Polynomial sdE2n+2m(z,k2)

When ν=2n+2, m=0,1,,n, the Fourier series (29.6.23) terminates:

29.15.28 sdE2n+2m(z,k2)=dn(z,k)p=0nC2p+1cos((2p+1)ϕ).

In (29.15.2) replace αp, βp, and γp as in (29.6.26). Also replace (29.15.4), (29.15.5), (29.15.6) by

29.15.29 [C1,C3,,C2n+1]T,
29.15.30 (1-12k2)p=0nC2p+12-12k2(12C12+p=0n-1C2p+1C2p+3)=1,
29.15.31 p=0nC2p+1>0.

Then

29.15.32 aν2m+1(k2)=12(Hm+ν(ν+1)k2),

and (29.15.28) applies.

Polynomial cdE2n+2m(z,k2)

When ν=2n+2, m=0,1,,n, the Fourier series (29.6.38) terminates:

29.15.33 cdE2n+2m(z,k2)=dn(z,k)p=0nD2p+1sin((2p+1)ϕ).

In (29.15.2) replace αp, βp, and γp as in (29.6.41). Also replace (29.15.4), (29.15.5), (29.15.6) by

29.15.34 [D1,D3,,D2n+1]T,
29.15.35 (1-12k2)p=0nD2p+12+12k2(12D12-p=0n-1D2p+1D2p+3)=1,
29.15.36 p=0n(2p+1)D2p+1>0.

Then

29.15.37 bν2m+1(k2)=12(Hm+ν(ν+1)k2),

and (29.15.33) applies.

Polynomial scdE2n+3m(z,k2)

When ν=2n+3, m=0,1,,n, the Fourier series (29.6.53) terminates:

29.15.38 scdE2n+3m(z,k2)=dn(z,k)p=0nD2p+2sin((2p+2)ϕ).

In (29.15.2) replace αp, βp, and γp as in (29.6.56). Also replace (29.15.4), (29.15.5), (29.15.6) by

29.15.39 [D2,D4,,D2n+2]T,
29.15.40 (1-12k2)p=0nD2p+22-12k2p=1nD2pD2p+2=1,
29.15.41 p=0n(2p+2)D2p+2>0.

Then

29.15.42 bν2m+2(k2)=12(Hm+ν(ν+1)k2),

and (29.15.38) applies.

§29.15(ii) Chebyshev Series

The Chebyshev polynomial T of the first kind (§18.3) satisfies cos(pϕ)=Tp(cosϕ). Since (29.2.5) implies that cosϕ=sn(z,k), (29.15.1) can be rewritten in the form

29.15.43 uE2nm(z,k2)=12A0+p=1nA2pT2p(sn(z,k)).

This determines the polynomial P of degree n for which uE2nm(z,k2)=P(sn2(z,k)); compare Table 29.12.1. The set of coefficients of this polynomial (without normalization) can also be found directly as an eigenvector of an (n+1)×(n+1) tridiagonal matrix; see Arscott and Khabaza (1962).

Using also sin((p+1)ϕ)=(sinϕ)Up(cosϕ), with U denoting the Chebyshev polynomial of the second kind (§18.3), we obtain

29.15.44 sE2n+1m(z,k2) =p=0nA2p+1T2p+1(sn(z,k)),
29.15.45 cE2n+1m(z,k2) =cn(z,k)p=0nB2p+1U2p(sn(z,k)),
29.15.46 dE2n+1m(z,k2) =dn(z,k)(12C0+p=1nC2pT2p(sn(z,k))),
29.15.47 scE2n+2m(z,k2) =cn(z,k)p=0nB2p+2U2p+1(sn(z,k)),
29.15.48 sdE2n+2m(z,k2) =dn(z,k)p=0nC2p+1T2p+1(sn(z,k)),
29.15.49 cdE2n+2m(z,k2) =cn(z,k)dn(z,k)p=0nD2p+1U2p(sn(z,k)),
29.15.50 scdE2n+3m(z,k2) =cn(z,k)dn(z,k)p=0nD2p+2U2p+1(sn(z,k)).

For explicit formulas for Lamé polynomials of low degree, see Arscott (1964b, p. 205).