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29 Lamé FunctionsApplications

§29.18 Mathematical Applications

Contents

§29.18(i) Sphero-Conal Coordinates

The wave equation

29.18.1 2u+ω2u=0,

when transformed to sphero-conal coordinates r,β,γ:

29.18.2 x =krsn(β,k)sn(γ,k),
y =ikkrcn(β,k)cn(γ,k),
z =1krdn(β,k)dn(γ,k),

with

29.18.3 r 0,
β =K+iβ,
0 β
2K,
0 γ4K,

admits solutions

29.18.4 u(r,β,γ)=u1(r)u2(β)u3(γ),

where u1, u2, u3 satisfy the differential equations

29.18.5 ddr(r2du1dr)+(ω2r2-ν(ν+1))u1 =0,
29.18.6 d2u2dβ2+(h-ν(ν+1)k2sn2(β,k))u2 =0,
29.18.7 d2u3dγ2+(h-ν(ν+1)k2sn2(γ,k))u3 =0,

with separation constants h and ν. (29.18.5) is the differential equation of spherical Bessel functions (§10.47(i)), and (29.18.6), (29.18.7) agree with the Lamé equation (29.2.1).

§29.18(ii) Ellipsoidal Coordinates

The wave equation (29.18.1), when transformed to ellipsoidal coordinates α,β,γ:

29.18.8 x =ksn(α,k)sn(β,k)sn(γ,k),
y =-kkcn(α,k)cn(β,k)cn(γ,k),
z =ikkdn(α,k)dn(β,k)dn(γ,k),

with

29.18.9 α =K+iK-α,
0α<K,
β =K+iβ,
0β2K,0γ4K,

admits solutions

29.18.10 u(α,β,γ)=u1(α)u2(β)u3(γ),

where u1, u2, u3 each satisfy the Lamé wave equation (29.11.1).

§29.18(iii) Spherical and Ellipsoidal Harmonics

See Erdélyi et al. (1955, §15.7).

§29.18(iv) Other Applications

Triebel (1965) gives applications of Lamé functions to the theory of conformal mappings. Patera and Winternitz (1973) finds bases for the rotation group.