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  "body": "\n## Summary\n\nA hypersingular integral equation is an integral equation with a very singular kernel.\nIts integral exists neither as an ordinary integral nor as a [[Cauchy principal value|0SYR10HD4R]].\nThe typical kernel on an interval is $1/(t - x)^2$, and the integral is read as a [[Hadamard finite part|G7VE41JHHH]].\nSuch equations arise in crack problems of elasticity and in the scattering of waves by thin screens.\nThey also arise in boundary element methods that use the normal derivative of a double layer potential.\n\n## Explanations\n\n### L1 human\n\nIn a Cauchy singular equation the kernel behaves like $1/(t - x)$, and the two sides of the singularity cancel.\nIn a hypersingular equation the kernel behaves like $1/(t - x)^2$, which is positive on both sides, so nothing cancels.\nOne gives the integral a meaning by removing its divergent part.\nA useful picture: the hypersingular operator is the derivative of a Cauchy singular operator.\nFor example, the finite part of $\\frac{1}{\\pi}\\int_{-1}^{1} \\frac{\\sqrt{1 - t^2}}{(t - x)^2}\\, dt$ equals $-1$ for every $x$ in $(-1, 1)$.\n",
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