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29 Lamé FunctionsLamé Polynomials

§29.17 Other Solutions

  1. §29.17(i) Second Solution
  2. §29.17(ii) Algebraic Lamé Functions
  3. §29.17(iii) Lamé–Wangerin Functions

§29.17(i) Second Solution

If (29.2.1) admits a Lamé polynomial solution E, then a second linearly independent solution F is given by

29.17.1 F(z)=E(z)iKzdu(E(u))2.

For properties of these solutions see Arscott (1964b, §9.7), Erdélyi et al. (1955, §15.5.1), Shail (1980), and Sleeman (1966b).

§29.17(ii) Algebraic Lamé Functions

Algebraic Lamé functions are solutions of (29.2.1) when ν is half an odd integer. They are algebraic functions of sn(z,k), cn(z,k), and dn(z,k), and have primitive period 8K. See Erdélyi (1941c), Ince (1940b), and Lambe (1952).

§29.17(iii) Lamé–Wangerin Functions

Lamé–Wangerin functions are solutions of (29.2.1) with the property that (sn(z,k))1/2w(z) is bounded on the line segment from iK to 2K+iK. See Erdélyi et al. (1955, §15.6).