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  "body": "\n## Summary\n\nEach [[integral transform|YCK709QB06]] has the form $(Tf)(u) = \\int_{t_1}^{t_2} f(t)\\, K(t, u)\\, dt$, and it is fixed by its [[kernel|9M0NNNFJGM]] and its interval of integration.\nThe table below lists common transforms with one standard normalisation; books differ in constants and signs.\n\n| Transform | Kernel $K(t, u)$ | Interval |\n|---|---|---|\n| [[Laplace transform|0V49JEZW06]] | $e^{-ut}$ | $(0, \\infty)$ |\n| [[Two-sided Laplace transform|6J2AGD0BYY]] | $e^{-ut}$ | $(-\\infty, \\infty)$ |\n| [[Fourier transform|DPVM34ADZV]] | $e^{-2\\pi i u t}$ | $(-\\infty, \\infty)$ |\n| [[Mellin transform|N5PVGRJE1T]] | $t^{u - 1}$ | $(0, \\infty)$ |\n| [[Hankel transform|2P9YWDGKC4]] | $t\\, J_\\nu(ut)$ | $(0, \\infty)$ |\n| [[Stieltjes transform|RG1HFAYD98]] | $1/(u + t)$ | $(0, \\infty)$ |\n| [[Hilbert transform|NSCEBXM7ZM]] | $\\frac{1}{\\pi}\\frac{1}{u - t}$ (principal value) | $(-\\infty, \\infty)$ |\n| [[Weierstrass transform|5HYB6DDZH8]] | $\\frac{1}{\\sqrt{4\\pi}} e^{-(u - t)^2/4}$ | $(-\\infty, \\infty)$ |\n| [[Hartley transform|WEHYECYAVJ]] | $\\frac{1}{\\sqrt{2\\pi}}(\\cos ut + \\sin ut)$ | $(-\\infty, \\infty)$ |\n\nHere $J_\\nu$ is the Bessel function of the first kind of order $\\nu$.\nSome transforms do not fit this one-line form.\nThe [[Radon transform|VJNZJ352PC]] integrates over lines, and the [[Abel transform|QMW1MWQH6J]] has a variable domain of integration.\n\n## Explanations\n\n### L1 human\n\nAll these transforms follow one recipe: multiply $f$ by a kernel and integrate.\nWhat changes is the kernel and the range of integration.\nExponential kernels suit differential equations, power kernels $t^{u-1}$ suit problems with scaling, and Bessel kernels suit problems with circular symmetry.\nFor example, the Mellin transform of $e^{-t}$ is the gamma function $\\Gamma(u)$.\n",
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