| 翻訳と辞書 |
| Hartogs' extension theorem ： ウィキペディア英語版 |
Hartogs' extension theorem
In mathematics, precisely in the theory of functions of several complex variables, Hartogs' extension theorem is a statement about the singularities of holomorphic functions of several variables. Informally, it states that the support of the singularities of such functions cannot be compact, therefore the singular set of a function of several complex variables must (loosely speaking) 'go off to infinity' in some direction. More precisely, it shows that the concept of isolated singularity and removable singularity coincide for analytic functions of complex variables. A first version of this theorem was proved by Friedrich Hartogs,〔See the original paper of and its description in various historical surveys by , and . In particular, in this last reference on p. 132, the Author explicitly writes :-"''As it is pointed out in the title of , and as the reader shall soon see, the key tool in the proof is the Cauchy integral formula''".〕 and as such it is known also as Hartogs' lemma and Hartogs' principle: in earlier Soviet literature,〔See for example , which refers the reader to the book of for a proof (however, in the former reference it is incorrectly stated that the proof is on page 324).〕 it is also called Osgood-Brown theorem, acknowledging later work by Arthur Barton Brown and William Fogg Osgood.〔See and .〕 This property of holomorphic functions of several variables is also called Hartogs' phenomenon: however, the locution "Hartogs' phenomenon" is also used to identify the property of solutions of systems of partial differential or convolution equations satisfying Hartogs type theorems.〔See and .〕
==Historical note==
The original proof was given by Friedrich Hartogs in 1906, using Cauchy's integral formula for functions of several complex variables.〔 Today, usual proofs rely on either Bochner–Martinelli–Koppelman formula or the solution of the inhomogeneous Cauchy–Riemann equations with compact support. The latter approach is due to Leon Ehrenpreis who initiated it in the paper . Yet another very simple proof of this result was given by Gaetano Fichera in the paper , by using his solution of the Dirichlet problem for holomorphic functions of several variables and the related concept of CR-function:〔Fichera's prof as well as his epoch making paper seem to have been overlooked by many specialists of the theory of functions of several complex variables: see for the correct attribution of many important theorems in this field.〕 later he extended the theorem to a certain class of partial differential operators in the paper , and his ideas were later further explored by Giuliano Bratti.〔See .〕 Also the Japanese school of the theory of partial differential operators worked much on this topic, with notable contributions by Akira Kaneko.〔See his paper and the references therein.〕 Their approach is to use Ehrenpreis' fundamental principle.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
■ウィキペディアで「Hartogs' extension theorem」の詳細全文を読む
スポンサード リンク
| 翻訳と辞書 : 翻訳のためのインターネットリソース |
Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.
|
|