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29 Lamé FunctionsLamé Functions

§29.10 Lamé Functions with Imaginary Periods

The substitutions

29.10.1 h=ν(ν+1)-h,
29.10.2 z=i(z-K-iK),

transform (29.2.1) into

29.10.3 d2wdz2+(h-ν(ν+1)k2sn2(z,k))w=0.

In consequence, the functions

29.10.4 Ecν2m(i(z-K-iK),k2),
Ecν2m+1(i(z-K-iK),k2),
Esν2m+1(i(z-K-iK),k2),
Esν2m+2(i(z-K-iK),k2),

are solutions of (29.2.1). The first and the fourth functions have period 2iK; the second and the third have period 4iK.

For these results and further information see Erdélyi et al. (1955, §15.5.2).