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  "body": "\n## Statement\n\nLet $C \\subseteq \\R^n$ be a nonempty open [[convex cone|6VE4KA12VD]] with vertex $0$, let $E \\subseteq \\R^n$ be a nonempty [[open set|02BH3MCBQB]], and let $\\delta \u003e 0$.\nWrite $\\R^n \\subseteq \\C^n$ as the set of points with zero imaginary part, and define the two wedges\n$$W_\\pm = \\set{x + iy \\in \\C^n : x \\in E, \\ \\pm y \\in C, \\ \\abs{y} \u003c \\delta}.$$\nLet $f_+$ and $f_-$ be [[holomorphic functions|507C34W0ZT]] on $W_+$ and $W_-$.\nSuppose that $f_+$ extends continuously to $W_+ \\cup E$, that $f_-$ extends continuously to $W_- \\cup E$, and that the two extensions agree on $E$.\nThen there are an open set $G \\subseteq \\C^n$ with $E \\subseteq G$ and a holomorphic function $F$ on $G$ such that $F = f_+$ on $G \\cap W_+$ and $F = f_-$ on $G \\cap W_-$.\n\n## Remarks\n\nFor $n = 1$ the wedges are a rectangle above and a rectangle below the interval $E$.\nThen the theorem follows from Morera's theorem: the glued function is continuous, and its integrals over small triangles vanish.\nIn several variables the edge $E$ has only real dimension $n$ inside $\\C^n$, and a direct Morera argument is not available.\nProofs use Cauchy integrals over suitable families of analytic discs, or Fourier analysis.\nThe theorem also holds when the continuity hypothesis is replaced by the condition that $f_+$ and $f_-$ have the same boundary value on $E$ in the sense of distributions.\n\nThe agreement on $E$ is necessary: $f_+ = 0$ and $f_- = 1$ satisfy all other hypotheses but cannot be glued to one holomorphic function.\n\n## Explanations\n\n### L1 human\n\nTake two wedge-shaped regions of $\\C^n$ that meet only along a common real edge.\nSuppose a holomorphic function on each wedge has boundary values on the edge, and the two sets of boundary values agree.\nThen the two functions are pieces of a single holomorphic function on a full neighbourhood of the edge.\nThe edge acts like a seam that disappears once the two sides match.\n\n### L2 human\n\nIn one variable this is the familiar gluing of holomorphic functions across a real interval.\nIn several variables it is surprising, because the edge has only half the real dimension of the space, yet it still suffices for gluing.\nThe theorem arose in quantum field theory, where it is used to prove analyticity properties of correlation functions.\nIt is also a basic tool in the theory of hyperfunctions, where generalised functions on $\\R^n$ are represented by boundary values of holomorphic functions on wedges.\n",
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