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29 Lamé FunctionsLamé Functions

§29.8 Integral Equations

Let w(z) be any solution of (29.2.1) of period 4K, w2(z) be a linearly independent solution, and 𝒲{w,w2} denote their Wronskian. Also let x be defined by

29.8.1 x=k2sn(z,k)sn(z1,k)sn(z2,k)sn(z3,k)-k2k2cn(z,k)cn(z1,k)cn(z2,k)cn(z3,k)+1k2dn(z,k)dn(z1,k)dn(z2,k)dn(z3,k),

where z,z1,z2,z3 are real, and sn, cn, dn are the Jacobian elliptic functions (§22.2). Then

29.8.2 μw(z1)w(z2)w(z3)=-2K2KPν(x)w(z)dz,

where Pν(x) is the Ferrers function of the first kind (§14.3(i)),

29.8.3 μ=2στ𝒲{w,w2},

and σ (= ±1) and τ are determined by

29.8.4 w(z+2K) =σw(z),
w2(z+2K) =τw(z)+σw2(z).

A special case of (29.8.2) is

29.8.5 Ecν2m(z1,k2)w2(K)-w2(-K)dw2(z)/dz|z=0=-KKPν(y)Ecν2m(z,k2)dz,


29.8.6 y=1kdn(z,k)dn(z1,k).

Others are:

29.8.7 Ecν2m+1(z1,k2)w2(K)+w2(-K)w2(0)=-k2sn(z1,k)-KKsn(z,k)dPν(y)dyEcν2m+1(z,k2)dz,
29.8.8 Esν2m+1(z1,k2)dw2(z)/dz|z=K+dw2(z)/dz|z=-Kdw2(z)/dz|z=0=k2kcn(z1,k)-KKcn(z,k)dPν(y)dyEsν2m+1(z,k2)dz,


29.8.9 Esν2m+2(z1,k2)dw2(z)/dz|z=K-dw2(z)/dz|z=-Kw2(0)=-k4ksn(z1,k)cn(z1,k)-KKsn(z,k)cn(z,k)d2Pν(y)dy2Esν2m+2(z,k2)dz.

For further integral equations see Arscott (1964a), Erdélyi et al. (1955, §15.5.3), Shail (1980), Sleeman (1968a), and Volkmer (1982, 1983, 1984).