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  "body": "\n## Summary\n\nThe Mehler–Fock transform is an [[integral transform|YCK709QB06]] on the half-line $[1, \\infty)$ whose kernel is a conical [[Legendre function|69XMERM6KJ]] $P_{-1/2 + i\\tau}(x)$.\nIt is an index transform: the transform variable $\\tau$ sits in the degree of the Legendre function, not in its argument.\nIt plays for the Legendre operator the role that the Fourier transform plays for $d^2/dx^2$.\nIt is used to solve boundary value problems for cones and in harmonic analysis on the hyperbolic plane.\n\n## Explanations\n\n### L1 human\n\nThe functions $P_{-1/2 + i\\tau}$, one for each $\\tau \\ge 0$, are the basic waves of the Legendre differential operator on $[1, \\infty)$.\nThe transform measures how much of each wave a function contains, as a Fourier transform measures frequencies.\nThe inverse transform adds the waves back with the right weights.\nFor example, the transform of the function equal to $1$ on $[1, 2]$ and $0$ elsewhere is $\\int_1^2 P_{-1/2 + i\\tau}(x)\\, dx$, a smooth function of $\\tau$.\n",
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