Lamé functions are solutions of Lamé's differential equation. In the Jacobi form,
![w^('')+[h-nu(nu+1)k^2sn^2(z,k)]w=0,](/images/equations/LameFunctions/NumberedEquation1.svg) |
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.
