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  "body": "\n## Statement\n\nLet $c \\in \\R$ and let $\\varphi$ be a complex function on the line $\\operatorname{Re} s = c$ such that $y \\mapsto \\varphi(c + iy)$ is [[integrable|X7GTF3D98F]] on $\\R$.\nDefine $\\psi \\colon \\R \\to \\C$ by $\\psi(t) = \\varphi(c + 2\\pi i t)$.\nThen for every $x \u003e 0$,\n$$\\mathcal{M}^{-1}_c \\varphi(x) = x^{-c} \\, \\check{\\psi}(-\\log x),$$\nwhere $\\mathcal{M}^{-1}_c$ is the [[inverse Mellin transform|CHESR1W1CY]] along $\\operatorname{Re} s = c$, and $\\check{\\psi}(u) = \\int_{\\R} \\psi(t) e^{2\\pi i u t} \\, dt$ is the [[inverse Fourier transform|JB8QMT8DCH]].\n\n## Remarks\n\nSubstitute $y = 2\\pi t$ in the definition:\n$$\\mathcal{M}^{-1}_c \\varphi(x) = \\frac{1}{2\\pi} \\int_{\\R} x^{-c - iy} \\varphi(c + iy) \\, dy = x^{-c} \\int_{\\R} e^{-2\\pi i t \\log x} \\, \\psi(t) \\, dt,$$\nand the last integral is $\\check{\\psi}(-\\log x)$.\nThe function $\\psi$ is integrable because $\\int \\abs{\\psi(t)} \\, dt = \\frac{1}{2\\pi} \\int \\abs{\\varphi(c + iy)} \\, dy$.\nExample: $\\varphi = \\Gamma$ and $c = 1$ give $x^{-1} \\check{\\psi}(-\\log x) = e^{-x}$, that is, $\\check{\\psi}(u) = e^{-u} e^{-e^{-u}}$ for $\\psi(t) = \\Gamma(1 + 2\\pi i t)$.\nCombined with the statement that the Mellin transform is a Fourier transform in logarithmic variables, this reduces the Mellin inversion theorem to the [[Fourier inversion theorem|T4Z8YQZKZV]].\n\n## Explanations\n\n### L1 human\n\nThe inverse Mellin transform is the inverse Fourier transform in disguise.\nOne rescales the vertical line so that it becomes the real line, takes the inverse Fourier transform, reads it at the point $-\\log x$, and multiplies by the weight $x^{-c}$.\n\n### L2 human\n\nThis identity is the inverse half of the dictionary between Mellin and Fourier analysis on $\\R$.\nIt turns questions about the inverse Mellin integral, such as continuity in $x$, decay or behaviour under smoothing, into the corresponding questions for the inverse Fourier transform.\nFor instance, it shows at once that $x \\mapsto x^{c} \\mathcal{M}^{-1}_c \\varphi(x)$ is bounded and continuous on $(0, \\infty)$.\n",
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