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  "body": "\n## Definition\n\nFor real $\\tau \\ge 0$, let $P_{-1/2 + i\\tau}$ be the conical [[Legendre function|69XMERM6KJ]] of the first kind on $[1, \\infty)$.\nLet $f \\colon [1, \\infty) \\to \\C$ be a function such that the integral below converges for every $\\tau \\ge 0$.\nThe *Mehler–Fock transform* of $f$ is the function\n$$F(\\tau) = \\int_1^\\infty f(x)\\, P_{-1/2 + i\\tau}(x)\\, dx, \\qquad \\tau \\ge 0.$$\n\n## Remarks\n\nUnder suitable conditions on $f$, the inversion formula is\n$$f(x) = \\int_0^\\infty \\tau \\tanh(\\pi \\tau)\\, P_{-1/2 + i\\tau}(x)\\, F(\\tau)\\, d\\tau, \\qquad x \u003e 1.$$\nThe function $P_{-1/2 + i\\tau}(x)$ is real for real $\\tau$ and $x \\ge 1$, and it decays like $x^{-1/2}$ up to a logarithmic factor as $x \\to \\infty$.\n\nExample: if $f$ is integrable and vanishes outside $[1, 2]$, then the integral is over a compact interval of a continuous kernel, so $F(\\tau)$ is defined for every $\\tau \\ge 0$.\n\nNon-example: $f(x) = x$ has no Mehler–Fock transform, since $x P_{-1/2 + i\\tau}(x)$ grows like $x^{1/2}$ up to a logarithmic factor and is not integrable on $[1, \\infty)$.\n\nThe substitution $x = \\cosh t$ turns the transform into an integral over $t \\in [0, \\infty)$, the form used on the hyperbolic plane.\n",
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