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  "body": "\n## Summary\n\nThe Mellin transform method solves integral equations on $(0, \\infty)$ whose kernel depends only on the ratio $x / t$ of the variables.\nSuch an integral is a [[Mellin convolution|A2PMJG7J68]], and the [[Mellin transform|N5PVGRJE1T]] turns it into a product.\nThe integral equation becomes an algebraic equation in a vertical strip of the complex plane, and the [[Mellin inversion theorem|XJWC13NY9E]] brings the solution back.\nThe method covers all kernels that are homogeneous of degree $-1$, such as $\\frac{1}{x + t}$.\nSuch kernels arise in problems with scaling symmetry, such as wedges, cones and cracks.\n\n## Explanations\n\n### L1 human\n\nThe [[Fourier transform method|NRG03YP2PW]] works when the kernel is unchanged by shifts $x \\mapsto x + c$.\nThe Mellin method works when the kernel is unchanged by stretches $x \\mapsto c x$.\nThe substitution $x = e^{u}$ turns stretches into shifts, so the Mellin method is the Fourier method in logarithmic coordinates.\nFor example, $\\int_0^\\infty \\frac{\\varphi(t)}{x + t} \\, dt = f(x)$ has a kernel that depends only on $x / t$ after taking out a factor $\\frac{1}{t}$.\nAfter the Mellin transform it reads $\\frac{\\pi}{\\sin \\pi s} \\Phi(s) = F(s)$, which is solved by division.\n",
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