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  "body": "\n## Summary\n\nA Lamé function is a solution of Lamé's equation $w'' + \\left( h - \\nu(\\nu + 1) k^2 \\operatorname{sn}^2(z, k) \\right) w = 0$, where $\\operatorname{sn}$ is a Jacobi [[elliptic function|FF9ZPCZ59K]].\nThe equation arises when [[Laplace's equation|S8RG6N1PRT]] is separated in ellipsoidal coordinates, and its solutions are also called ellipsoidal harmonics.\nWhen $\\nu = n$ is a nonnegative integer, special values of $h$ give Lamé polynomials, which play for ellipsoids the role that Legendre polynomials play for spheres.\nThe equation is a special case of Heun's equation and a periodic Schrödinger equation with an elliptic potential.\n\n## Explanations\n\n### L1 human\n\nLamé's equation is like the equation $w'' + h w = 0$ of a vibrating string, but with an extra term that oscillates periodically along $z$.\nThe oscillating term is a square of an elliptic sine, which is periodic like the sine but has a shape controlled by a modulus $k$.\nFor degree $\\nu = 1$ the three functions $\\operatorname{sn} z$, $\\operatorname{cn} z$ and $\\operatorname{dn} z$ are solutions, for $h = 1 + k^2$, $h = 1$ and $h = k^2$ respectively.\nThese are the simplest Lamé polynomials.\n",
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