Lamé functions are solutions of Lamé's differential equation. In the Jacobi form,

where primes denote derivatives with respect to and is a Jacobi elliptic function of elliptic modulus . For real and , the coefficient of is -periodic, where is the complete elliptic integral of the first kind. Solutions that are periodic with period or occur at discrete eigenvalues of (Erdélyi et al. 1955, §15.5.1; NIST DLMF, §29.3(i)).

The two families reflect the symmetry of the coefficient of about . An even solution about that point has , and an odd solution has . Both families admit even and odd solutions about . Composing the two reflections translates by . Equal reflection parities give period , while opposite parities give a sign change after and period (NIST DLMF, Table 29.3.2).

The Wolfram Language functions LameC[nu, j, z, m] and LameS[nu, j, z, m] give the families even and odd about , respectively, for real and elliptic parameter with . The index ranges over the nonnegative integers for LameC and the positive integers for LameS. The corresponding characteristic values are given by LameEigenvalueA and LameEigenvalueB, respectively. In ellipsoidal coordinates, the names Lamé functions of the first and second kinds refer to the ellipsoidal harmonic of the first kind and the ellipsoidal harmonic of the second kind, respectively.

See also

Ellipsoidal Harmonic of the First Kind, Ellipsoidal Harmonic of the Second Kind, Lamé's Differential Equation

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References

Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 3. New York: McGraw-Hill, Ch. 15, 1955.National Institute of Standards and Technology. "Eigenvalues." §29.3(i) in Digital Library of Mathematical Functions. https://dlmf.nist.gov/29.3.i.

Cite this as:

Weisstein, Eric W. "Lamé Functions." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LameFunctions.html

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