{"history":[],"article":{"id":"61cf689c-8901-42af-a2d8-009135200d21","title":"Edge-of-the-wedge theorem","slug":"edge-of-the-wedge-theorem","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), Bogoliubov's **edge-of-the-wedge theorem** implies that [holomorphic functions](https://bliptext.com/articles/holomorphic-function) on two \"wedges\" with an \"edge\" in common are [analytic continuations](https://bliptext.com/articles/analytic-continuation) of each other provided they both give the same continuous function on the edge.\nIt is used in [quantum field theory](https://bliptext.com/articles/quantum-field-theory) to construct the [analytic continuation](https://bliptext.com/articles/analytic-continuation) of [Wightman functions](https://bliptext.com/articles/wightman-function).\nThe formulation and the first proof of the theorem were presented by [Nikolay Bogoliubov](https://bliptext.com/articles/nikolay-bogoliubov) at the International Conference on Theoretical Physics, Seattle, USA (September, 1956) and also published in the book Problems in the Theory of Dispersion Relations.\nFurther proofs and generalizations of the theorem were given by [Res Jost](https://bliptext.com/articles/res-jost) and [Harry Lehmann](https://bliptext.com/articles/harry-lehmann) (1957), [Freeman Dyson](https://bliptext.com/articles/freeman-dyson) (1958), H. Epstein (1960), and by other researchers.\n\n# The one-dimensional case\n\n## Continuous boundary values\nIn one dimension, a simple case of the edge-of-the-wedge theorem can be stated as follows.\nIn this example, the two wedges are the upper half-plane and the lower half plane, and their common edge is the [real axis](https://bliptext.com/articles/real-axis).\nThis result can be proved from [Morera's theorem](https://bliptext.com/articles/morera-s-theorem).\nIndeed, a function is holomorphic provided its integral round any contour vanishes; a contour which crosses the real axis can be broken up into contours in the upper and lower half-planes and the integral round these vanishes by hypothesis.\n\n## Distributional boundary values on a circle\nThe more general case is phrased in terms of distributions.\nThis is technically simplest in the case where the common boundary is the unit circle |z|=1 in the complex plane.\nIn that case holomorphic functions f, g in the regions r<|z|<1 and 1<|z|

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