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  "body": "\n## Definition\n\nLet $a, b \\colon (-1, 1) \\to \\R$ and $f \\colon (-1, 1) \\to \\R$ be given functions, and let $K$ be a [[kernel|GSMBXT7FPK]] on $(-1, 1)^2$ that is integrable in $t$ for each $x$.\nA *hypersingular integral equation* on $(-1, 1)$ for an unknown $\\varphi$ is\n$$a(x)\\, \\varphi(x) + \\frac{b(x)}{\\pi}\\, \\mathrm{f.p.}\\!\\int_{-1}^{1} \\frac{\\varphi(t)}{(t - x)^2}\\, dt + \\int_{-1}^{1} K(x, t)\\, \\varphi(t)\\, dt = f(x), \\quad -1 \u003c x \u003c 1,$$\nwhere $\\mathrm{f.p.}$ denotes the [[Hadamard finite part|G7VE41JHHH]] and $b$ does not vanish identically.\nThe unknown $\\varphi$ is sought among functions for which the finite part exists, for instance functions whose derivative is Hölder continuous on each compact subinterval.\nIf $a = 0$, the equation is of the *first kind*.\n\n## Remarks\n\nMore generally, an integral equation on a domain or surface of dimension $m$ is called hypersingular when its kernel grows like $\\abs{x - t}^{-m-1}$ or faster near the diagonal.\n\nExample: the first kind equation $\\frac{1}{\\pi}\\, \\mathrm{f.p.}\\!\\int_{-1}^{1} \\frac{\\varphi(t)}{(t - x)^2}\\, dt = -1$ has the solution $\\varphi(t) = \\sqrt{1 - t^2}$.\nCheck: the finite part is the $x$-derivative of $\\frac{1}{\\pi}\\, \\mathrm{PV}\\!\\int_{-1}^{1} \\frac{\\sqrt{1 - t^2}}{t - x}\\, dt = -x$, which is $-1$.\nHere $\\mathrm{PV}$ is the [[Cauchy principal value|0SYR10HD4R]].\nEquations of this form, with the condition $\\varphi(\\pm 1) = 0$, describe the opening of a straight crack under uniform pressure.\n\nNon-example: $\\frac{1}{\\pi}\\, \\mathrm{PV}\\!\\int_{-1}^{1} \\frac{\\varphi(t)}{t - x}\\, dt = f(x)$ has a Cauchy kernel of order $1$, so it is a singular but not a hypersingular equation.\n\nThe finite part is the derivative of a principal value.\nSo, when $\\varphi(\\pm 1) = 0$, an integration by parts turns a hypersingular equation of the first kind for $\\varphi$ into a Cauchy singular equation for $\\varphi'$.\n",
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