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29 Lamé FunctionsLamé Functions

§29.6 Fourier Series

Contents

§29.6(i) Function Ecν2m(z,k2)

With ϕ=12π-am(z,k), as in (29.2.5), we have

29.6.1 Ecν2m(z,k2)=12A0+p=1A2pcos(2pϕ).

Here

29.6.2 H=2aν2m(k2)-ν(ν+1)k2,
29.6.3 (β0-H)A0+α0A2=0,
29.6.4 γpA2p-2+(βp-H)A2p+αpA2p+2=0,
p1,

with αp, βp, and γp as in (29.3.11) and (29.3.12), and

29.6.5 12A02+p=1A2p2=1,
29.6.6 12A0+p=1A2p>0.

When ν2n, where n is a nonnegative integer, it follows from §2.9(i) that for any value of H the system (29.6.4)–(29.6.6) has a unique recessive solution A0,A2,A4,; furthermore

29.6.7 limpA2p+2A2p=k2(1+k)2,
ν2n, or ν=2n and m>n.

In addition, if H satisfies (29.6.2), then (29.6.3) applies.

In the special case ν=2n, m=0,1,,n, there is a unique nontrivial solution with the property A2p=0, p=n+1,n+2,. This solution can be constructed from (29.6.4) by backward recursion, starting with A2n+2=0 and an arbitrary nonzero value of A2n, followed by normalization via (29.6.5) and (29.6.6). Consequently, Ecν2m(z,k2) reduces to a Lamé polynomial; compare §§29.12(i) and 29.15(i).

An alternative version of the Fourier series expansion (29.6.1) is given by

29.6.8 Ecν2m(z,k2)=dn(z,k)(12C0+p=1C2pcos(2pϕ)).

Here dn(z,k) is as in §22.2, and

29.6.9 (β0-H)C0+α0C2=0,
29.6.10 γpC2p-2+(βp-H)C2p+αpC2p+2=0,
p1,

with αp,βp, and γp now defined by

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

and

29.6.12 (1-12k2)(12C02+p=1C2p2)-12k2p=0C2pC2p+2=1,
29.6.13 12C0+p=1C2p>0,
29.6.14 limpC2p+2C2p=k2(1+k)2,
ν2n+1, or ν=2n+1 and m>n,
29.6.15 12A0C0+p=1A2pC2p=4π0K(Ecν2m(x,k2))2x.

§29.6(ii) Function Ecν2m+1(z,k2)

29.6.16 Ecν2m+1(z,k2)=p=0A2p+1cos((2p+1)ϕ).

Here

29.6.17 H=2aν2m+1(k2)-ν(ν+1)k2,
29.6.18 (β0-H)A1+α0A3=0,
29.6.19 γpA2p-1+(βp-H)A2p+1+αpA2p+3=0,
p1,

with αp, βp, and γp as in (29.3.13) and (29.3.14), and

29.6.20 p=0A2p+12=1,
29.6.21 p=0A2p+1>0,
29.6.22 limpA2p+1A2p-1=k2(1+k)2,
ν2n+1, or ν=2n+1 and m>n.

Also,

29.6.23 Ecν2m+1(z,k2)=dn(z,k)p=0C2p+1cos((2p+1)ϕ),

where

29.6.24 (β0-H)C1+α0C3=0,
29.6.25 γpC2p-1+(βp-H)C2p+1+αpC2p+3=0,
p1,

with

29.6.26 αp =12(ν-2p-1)(ν+2p+2)k2,
βp ={2-k2+12ν(ν+1)k2,p=0,(2p+1)2(2-k2),p1,
γp =12(ν-2p)(ν+2p+1)k2,

and

29.6.27 (1-12k2)p=0C2p+12-12k2(12C12+p=0C2p+1C2p+3)=1,
29.6.28 p=0C2p+1>0,
29.6.29 limpC2p+1C2p-1=k2(1+k)2,
ν2n+2, or ν=2n+2 and m>n,
29.6.30 p=0A2p+1C2p+1=4π0K(Ecν2m+1(x,k2))2x.

§29.6(iii) Function Esν2m+1(z,k2)

29.6.31 Esν2m+1(z,k2)=p=0B2p+1sin((2p+1)ϕ).

Here

29.6.32 H=2bν2m+1(k2)-ν(ν+1)k2,
29.6.33 (β0-H)B1+α0B3=0,
29.6.34 γpB2p-1+(βp-H)B2p+1+αpB2p+3=0,
p1,

with αp, βp, and γp as in (29.3.15), (29.3.16), and

29.6.35 p=0B2p+12=1,
29.6.36 p=0(2p+1)B2p+1>0,
29.6.37 limpB2p+1B2p-1=k2(1+k)2,
ν2n+1, or ν=2n+1 and m>n.

Also,

29.6.38 Esν2m+1(z,k2)=dn(z,k)p=0D2p+1sin((2p+1)ϕ),

where

29.6.39 (β0-H)D1+α0D3=0,
29.6.40 γpD2p-1+(βp-H)D2p+1+αpD2p+3=0,
p1,

with

29.6.41 αp =12(ν-2p-1)(ν+2p+2)k2,
βp ={2-k2-12ν(ν+1)k2,p=0,(2p+1)2(2-k2),p1,
γp =12(ν-2p)(ν+2p+1)k2,

and

29.6.42 (1-12k2)p=0D2p+12+12k2(12D12-p=0D2p+1D2p+3)=1,
29.6.43 p=0(2p+1)D2p+1>0,
29.6.44 limpD2p+1D2p-1=k2(1+k)2,
ν2n+2, or ν=2n+2 and m>n,
29.6.45 p=0B2p+1D2p+1=4π0K(Esν2m+1(x,k2))2x.

§29.6(iv) Function Esν2m+2(z,k2)

29.6.46 Esν2m+2(z,k2)=p=1B2psin(2pϕ).

Here

29.6.47 H=2bν2m+2(k2)-ν(ν+1)k2,
29.6.48 (β0-H)B2+α0B4=0,
29.6.49 γpB2p+(βp-H)B2p+2+αpB2p+4=0,
p1,

with αp, βp, and γp as in (29.3.17), and

29.6.50 p=1B2p2=1,
29.6.51 p=0(2p+2)B2p+2>0,
29.6.52 limpB2p+2B2p=k2(1+k)2,
ν2n+2, or ν=2n+2 and m>n.

Also,

29.6.53 Esν2m+2(z,k2)=dn(z,k)p=1D2psin(2pϕ),

where

29.6.54 (β0-H)D2+α0D4=0,
29.6.55 γpD2p+(βp-H)D2p+2+αpD2p+4=0,
p1,

with

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

and

29.6.57 (1-12k2)p=1D2p2-12k2p=1D2pD2p+2=1,
29.6.58 p=0(2p+2)D2p+2>0,
29.6.59 limpD2p+2D2p=k2(1+k)2,
ν2n+3, or ν=2n+3 and m>n,
29.6.60 p=1B2pD2p=4π0K(Esν2m+2(x,k2))2x.