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  "body": "\n## Definition\n\nLet $k \\in (0, 1)$, and let $\\operatorname{sn}(z, k)$, $\\operatorname{cn}(z, k)$ and $\\operatorname{dn}(z, k)$ be the Jacobi [[elliptic functions|FF9ZPCZ59K]] of modulus $k$.\nLet $\\nu$ and $h$ be complex numbers.\n*Lamé's equation* is the [[linear differential equation|0QQRP17Z7S]]\n$$\\frac{d^2 w}{dz^2} + \\left( h - \\nu(\\nu + 1) k^2 \\operatorname{sn}^2(z, k) \\right) w = 0.$$\nA *Lamé function* is a solution $w$ of Lamé's equation.\nLet $\\nu = n$ be a nonnegative integer.\nA *Lamé polynomial* of degree $n$ is a nonzero solution of the form $\\operatorname{sn}^{e_1} z\\, \\operatorname{cn}^{e_2} z\\, \\operatorname{dn}^{e_3} z \\cdot P(\\operatorname{sn}^2 z)$.\nHere $e_1, e_2, e_3 \\in \\set{0, 1}$ and $P$ is a polynomial.\n\n## Remarks\n\nExample: for $\\nu = 1$ and $h = 1 + k^2$, the function $w = \\operatorname{sn} z$ is a Lamé polynomial.\nCheck: from $\\operatorname{sn}' = \\operatorname{cn} \\operatorname{dn}$, $\\operatorname{cn}^2 = 1 - \\operatorname{sn}^2$ and $\\operatorname{dn}^2 = 1 - k^2 \\operatorname{sn}^2$, one gets $\\operatorname{sn}'' = -(1 + k^2) \\operatorname{sn} + 2k^2 \\operatorname{sn}^3$, so $w'' + (1 + k^2 - 2k^2 \\operatorname{sn}^2) w = 0$.\nIn the same way $\\operatorname{cn} z$ is a solution for $h = 1$ and $\\operatorname{dn} z$ for $h = k^2$.\nNon-example: for $\\nu = 1$ and $h = 0$, the function $\\operatorname{sn} z$ is not a solution, since the left side equals $-(1 + k^2) \\operatorname{sn} z$, which is not identically $0$.\nFor each nonnegative integer $n$ there are exactly $2n + 1$ values of $h$ that admit a Lamé polynomial of degree $n$.\nThis matches the dimension $2n + 1$ of the space of spherical harmonics of degree $n$.\nAn equivalent form of the equation uses the Weierstrass function $\\wp$ in place of $k^2 \\operatorname{sn}^2$, and in the variable $\\operatorname{sn}^2 z$ it becomes a special case of Heun's equation.\n",
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