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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Subharmonic function ： ウィキペディア英語版 Subharmonic function In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory. Intuitively, subharmonic functions are related to convex functions of one variable as follows. If the graph of a convex function and a line intersect at two points, then the graph of the convex function is ''below'' the line between those points. In the same way, if the values of a subharmonic function are no larger than the values of a harmonic function on the ''boundary'' of a ball, then the values of the subharmonic function are no larger than the values of the harmonic function also ''inside'' the ball. ''Superharmonic'' functions can be defined by the same description, only replacing "no larger" with "no smaller". Alternatively, a superharmonic function is just the negative of a subharmonic function, and for this reason any property of subharmonic functions can be easily transferred to superharmonic functions. ==Formal definition== Formally, the definition can be stated as follows. Let $G$ be a subset of the Euclidean space $\}\; \backslash cup\; \backslash$ be an upper semi-continuous function. Then, $\backslash varphi$ is called ''subharmonic'' if for any closed ball $\backslash overline$ of center $x$ and radius $r$ contained in $G$ and every real-valued continuous function $h$ on $\backslash overline$ that is harmonic in $B(x,r)$ and satisfies $\backslash varphi(y)\; \backslash leq\; h(y)$ for all $y$ on the boundary $\backslash partial\; B(x,r)$ of $B(x,r)$ we have $\backslash varphi(y)\; \backslash leq\; h(y)$ for all $y\; \backslash in\; B(x,r).$ Note that by the above, the function which is identically −∞ is subharmonic, but some authors exclude this function by definition. A function $u$ is called ''superharmonic'' if $-u$ is subharmonic. 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「Subharmonic function」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.