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  "body": "\n## Definition\n\nLet $\\varpi = 2\\int_0^1 dt/\\sqrt{1 - t^4}$ be the [[lemniscate constant|V5A2S8QYQQ]].\nThe map $x \\mapsto \\int_0^x dt/\\sqrt{1 - t^4}$ is a strictly increasing bijection from $[-1, 1]$ onto $[-\\varpi/2, \\varpi/2]$.\nThe *lemniscate sine* $\\operatorname{sl}$ is the inverse map on $[-\\varpi/2, \\varpi/2]$, extended to $\\R$ by $\\operatorname{sl}(x + \\varpi) = -\\operatorname{sl}(x)$.\nIt extends further to a meromorphic function on $\\C$ by [[analytic continuation|0M3YQJ81NB]].\nThe *lemniscate cosine* is $\\operatorname{cl}(z) = \\operatorname{sl}(\\varpi/2 - z)$.\n\n## Remarks\n\nThe function $\\operatorname{sl}$ is odd, satisfies $\\operatorname{sl}(0) = 0$, $\\operatorname{sl}(\\varpi/2) = 1$, and $(\\operatorname{sl}')^2 = 1 - \\operatorname{sl}^4$.\nIt is an [[elliptic function|FF9ZPCZ59K]] whose periods form the lattice spanned by $(1 + i)\\varpi$ and $(1 - i)\\varpi$; in particular $2\\varpi$ and $2i\\varpi$ are periods.\nIt satisfies $\\operatorname{sl}(iz) = i\\operatorname{sl}(z)$.\nThe two functions are linked by $\\operatorname{sl}^2 + \\operatorname{cl}^2 + \\operatorname{sl}^2\\operatorname{cl}^2 = 1$.\nExample: at $z = \\varpi/2$ one has $\\operatorname{sl} = 1$ and $\\operatorname{cl} = \\operatorname{sl}(0) = 0$, and indeed $1 + 0 + 0 = 1$.\nNon-example: the inverse of $\\int_0^x dt/\\sqrt{1 - t^2}$ is $\\sin$, which is not the lemniscate sine: it has the single period $2\\pi$ and satisfies $(\\sin')^2 = 1 - \\sin^2$ instead of $1 - \\operatorname{sl}^4$.\nIn terms of the [[Jacobi elliptic functions|P071B692JF]], $\\operatorname{sl}(z) = \\operatorname{sn}(z; k)$ with $k^2 = -1$, since this $\\operatorname{sn}$ solves the same equation $(y')^2 = (1 - y^2)(1 + y^2)$ with $y(0) = 0$, $y'(0) = 1$.\n",
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