The General Case
A wedge is a product of a cone with some set.
Let C be an open cone in the real vector space Rn, with vertex at the origin. Let E be an open subset of Rn, called the edge. Write W for the wedge in the complex vector space Cn, and write W' for the opposite wedge . Then the two wedges W and W' meet at the edge E, where we identify E with the product of E with the tip of the cone.
- Suppose that f is a continuous function on the union that is holomorphic on both the wedges W and W' . Then the edge-of-the-wedge theorem says that f is also holomorphic on E (or more precisely, it can be extended to a holomorphic function on a neighborhood of E).
The conditions for the theorem to be true can be weakened. It is not necessary to assume that f is defined on the whole of the wedges: it is enough to assume that it is defined near the edge. It is also not necessary to assume that f is defined or continuous on the edge: it is sufficient to assume that the functions defined on either of the wedges have the same distributional boundary values on the edge.
Read more about this topic: Edge-of-the-wedge Theorem
Famous quotes containing the words general and/or case:
“He who never sacrificed a present to a future good or a personal to a general one can speak of happiness only as the blind do of colors.”
—Olympia Brown (1835–1900)
“Oh, that I knew where I might find him, that I might come even to his dwelling! I would lay my case before him, and fill my mouth with arguments. I would learn what he would answer me, and understand what he would say to me. Would he contend with me in the greatness of his power? No; but he would give heed to me. There an upright person could reason with him, and I should be acquitted forever by my judge.”
—Bible: Hebrew, Job 23:3-7.
Job, of God.