§29.17 Other Solutions

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

29.17.1

Symbols:

: Legendre’s complementary complete elliptic integral of the first kind, : differential of , : integral, : complex variable and : real parameter

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

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

Lamé–Wangerin functions are solutions of (29.2.1) with the property that is bounded on the line segment from to . See Erdélyi et al. (1955, §15.6).