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29 Lamé FunctionsApplications

§29.18 Mathematical Applications

  1. §29.18(i) Sphero-Conal Coordinates
  2. §29.18(ii) Ellipsoidal Coordinates
  3. §29.18(iii) Spherical and Ellipsoidal Harmonics
  4. §29.18(iv) Other Applications

§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),


29.18.3 r 0,
β =K+iβ,
0 β2K,
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)+(ω2r2ν(ν+1))u1 =0,
29.18.6 d2u2dβ2+(hν(ν+1)k2sn2(β,k))u2 =0,
29.18.7 d2u3dγ2+(hν(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),


29.18.9 α =K+iKα,
β =K+iβ,

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.