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  "body": "\n## Definition\n\nLet $U \\subseteq \\R$ be an open set.\nLet $V \\subseteq \\C$ be an open set with $V \\cap \\R = U$; then $U$ is a closed subset of $V$.\nFor an open set $W \\subseteq \\C$, let $\\mathcal{O}(W)$ be the space of [[holomorphic functions|0NAHXR1FRH]] on $W$.\nThe space of *hyperfunctions* on $U$ is the quotient vector space\n$$\\mathcal{B}(U) = \\mathcal{O}(V \\setminus U) / \\mathcal{O}(V),$$\nwhere $\\mathcal{O}(V)$ is embedded in $\\mathcal{O}(V \\setminus U)$ by restriction.\nThe class of $F \\in \\mathcal{O}(V \\setminus U)$ is written $[F]$.\n\n## Remarks\n\nUp to canonical isomorphism, $\\mathcal{B}(U)$ does not depend on the choice of $V$.\nWrite $F_+$ and $F_-$ for the restrictions of $F$ to the parts of $V$ above and below the real axis.\nOne thinks of $[F]$ as $F_+(x + i0) - F_-(x - i0)$.\nWhen both boundary values exist as distributions, this difference is a distribution, and every distribution on $U$ arises this way.\n\nExample: on $U = \\R$ with $V = \\C$, let $F(z) = -1/(2\\pi i z)$ on $\\C \\setminus \\R$.\nThe boundary values give $F_+(x + i0) - F_-(x - i0) = -\\frac{1}{2\\pi i}\\left(\\frac{1}{x + i0} - \\frac{1}{x - i0}\\right)$, and this is the [[Dirac delta|0KPHENZ5S7]].\nSo $\\delta = [F]$.\nA function holomorphic on all of $V$ gives the zero hyperfunction.\n\nNon-example: for $U = (0,1)$, the set $V = \\C$ does not qualify, because $V \\cap \\R = \\R \\neq U$.\nWith that choice, $\\mathcal{O}(\\C \\setminus (0,1)) / \\mathcal{O}(\\C)$ would also see singularities at the end points $0$ and $1$, which lie outside $U$.\n\nIn several variables the definition uses relative sheaf cohomology of the sheaf of holomorphic functions instead of a quotient.\n",
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