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29 Lamé FunctionsLamé Functions

§29.7 Asymptotic Expansions

  1. §29.7(i) Eigenvalues
  2. §29.7(ii) Lamé Functions

§29.7(i) Eigenvalues

As ν,

29.7.1 aνm(k2)pκτ0τ1κ1τ2κ2,


29.7.2 κ =k(ν(ν+1))1/2,
p =2m+1,
29.7.3 τ0 =123(1+k2)(1+p2),
29.7.4 τ1 =p26((1+k2)2(p2+3)4k2(p2+5)).

The same Poincaré expansion holds for bνm+1(k2), since

29.7.5 bνm+1(k2)aνm(k2)=O(νm+32(1k1+k)ν),

See also Volkmer (2004b).

29.7.6 τ2=1210(1+k2)(1k2)2(5p4+34p2+9),
29.7.7 τ3=p214((1+k2)4(33p4+410p2+405)24k2(1+k2)2(7p4+90p2+95)+16k4(9p4+130p2+173)),
29.7.8 τ4=1216((1+k2)5(63p6+1260p4+2943p2+486)8k2(1+k2)3(49p6+1010p4+2493p2+432)+16k4(1+k2)(35p6+760p4+2043p2+378)).

Formulas for additional terms can be computed with the author’s Maple program LA5; see §29.22.

§29.7(ii) Lamé Functions

Müller (1966a, b) found three formal asymptotic expansions for a fundamental system of solutions of (29.2.1) (and (29.11.1)) as ν, one in terms of Jacobian elliptic functions and two in terms of Hermite polynomials. In Müller (1966c) it is shown how these expansions lead to asymptotic expansions for the Lamé functions 𝐸𝑐νm(z,k2) and 𝐸𝑠νm(z,k2). Weinstein and Keller (1985) give asymptotics for solutions of Hill’s equation (§28.29(i)) that are applicable to the Lamé equation.