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  "body": "\n## Summary\n\nThe Jacobi transform is a finite [[integral transform|YCK709QB06]] on $[-1, 1]$.\nIt sends a function to the sequence of its integrals against the [[Jacobi polynomials|TRD7DSF2KR]] $P_n^{(\\alpha, \\beta)}$ with their weight.\nIt is the coefficient map of the expansion of a function in Jacobi polynomials.\nIt turns the Jacobi differential operator into multiplication by $-n(n + \\alpha + \\beta + 1)$, so it is used to solve boundary value problems on an interval.\nThe Gegenbauer and Legendre transforms are special cases.\n\n## Explanations\n\n### L1 human\n\nJacobi polynomials are a family of polynomials that are orthogonal on $[-1, 1]$ for a weight that may favour one end of the interval.\nAny reasonable function on $[-1, 1]$ can be written as a sum of these polynomials.\nThe Jacobi transform records the numbers that tell how much of each polynomial is needed.\nFor example, the transform of $P_2^{(\\alpha, \\beta)}$ itself is zero at every $n$ except $n = 2$.\nA differential equation for the function becomes a simple algebraic equation for these numbers.\nThe same name is also used for a transform on the half-line with Jacobi functions as kernel; this node is about the finite transform.\n",
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