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  "body": "\n## Summary\n\nThe finite Hilbert transform sends a function $f$ on the interval $(-1, 1)$ to the [[Cauchy principal value|0SYR10HD4R]] integral $\\frac{1}{\\pi}\\, \\mathrm{PV}\\!\\int_{-1}^{1} \\frac{f(t)}{t - x}\\, dt$.\nUp to sign, it is the [[Hilbert transform|NSCEBXM7ZM]] of $f$ extended by zero, looked at only on the interval.\nIt is the singular part of most singular integral equations on an arc, such as the [[airfoil equation|W6Z6CWRE25]].\nUnlike the Hilbert transform on the line, it is not invertible on $L^p(-1, 1)$ for any $1 \u003c p \u003c \\infty$.\nFor $1 \u003c p \u003c 2$ it even sends a nonzero function to $0$.\n\n## Explanations\n\n### L1 human\n\nTake a function on a segment and average it against $1/(t - x)$, which is large and changes sign at $t = x$.\nThe two sides of the singularity nearly cancel, and the principal value keeps what is left.\nFor example, the transform of $\\sqrt{1 - t^2}$ is $-x$, and the transform of $1/\\sqrt{1 - t^2}$ is $0$.\nSo the transform kills a nonzero function, which is why equations with it need an extra condition to have one solution.\n",
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