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  "body": "\n## Summary\n\nThe inverse Mellin transform turns a function $\\varphi$ on a vertical line $\\operatorname{Re} s = c$ into the function $x \\mapsto \\frac{1}{2\\pi i} \\int_{c - i\\infty}^{c + i\\infty} x^{-s} \\varphi(s) \\, ds$ on $(0, \\infty)$.\nBy the [[Mellin inversion theorem|XJWC13NY9E]], it undoes the [[Mellin transform|N5PVGRJE1T]] under suitable hypotheses.\nIn logarithmic variables it is an [[inverse Fourier transform|JB8QMT8DCH]], and when $\\varphi$ is meromorphic it can often be computed with the [[residue theorem|032HKYS329]].\n\n## Explanations\n\n### L1 human\n\nThe Mellin transform breaks a function on $(0, \\infty)$ into the powers $x^{-s}$, with $s$ on a vertical line.\nThe inverse Mellin transform adds these powers back together, with the weights $\\varphi(s)$.\nFor example, with $\\varphi = \\Gamma$ and any $c \u003e 0$, the inverse transform gives back $e^{-x}$.\nMoving the line of integration to the left and collecting residues of $\\Gamma$ at $0, -1, -2, \\dots$ then gives the power series of $e^{-x}$.\n",
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