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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 Bounded type (mathematics) ： ウィキペディア英語版 Bounded type (mathematics) In mathematics, a function defined on a region of the complex plane is said to be of bounded type if it is analytic in the region and equal to the ratio of two analytic functions bounded in that region. But more generally, a function is of bounded type in a region $\backslash Omega$ if and only if $f$ is analytic on $\backslash Omega$ and $\backslash log^+|f(z)|$ has a harmonic majorant on $\backslash Omega,$ where $\backslash log^+(x)=\backslash max()$. Being the ratio of two bounded analytic functions is a sufficient condition for a function to be of bounded type (defined in terms of a harmonic majorant), and if $\backslash Omega$ is simply connected the condition is also necessary. The class of all such $f$ on $\backslash Omega$ is commonly denoted $N(\backslash Omega)$ and is sometimes called the ''Nevanlinna class'' for $\backslash Omega$. The Nevanlinna class includes all the Hardy classes. Functions of bounded type are not necessarily bounded, nor do they have a property called "type" which is bounded. The reason for the name is probably that when defined on a disc, the Nevanlinna characteristic (a function of distance from the centre of the disc) is bounded. ==Examples== Polynomials are of bounded type in any bounded region. They are also of bounded type in the upper half-plane (UHP), because a polynomial $f(z)$ of degree ''n'' can be expressed as a ratio of two analytic functions bounded in the UHP: :$f(z)=P(z)/Q(z)$ with :$P(z)=f(z)/(z+i)^n$ :$Q(z)=1/(z+i)^n.$ The inverse of a polynomial is of bounded type in a region so long as the polynomial has no root in the region, although it may have roots on the boundary of the region. If the polynomial has a root in the region, then the inverse is not considered to be of bounded type simply because it is not analytic at the root. The function $\backslash exp(aiz)$ is of bounded type in the UHP if and only if ''a'' is real. If ''a'' is positive the function itself is bounded in the UHP (so we can use $Q(z)=1$), and if ''a'' is negative then the function equals 1/Q(z) with $Q(z)=\backslash exp(|a|iz)$. Sine and cosine are of bounded type in the UHP. Indeed, :$\backslash sin(z)=P(z)/Q(z)$ with :$P(z)=\backslash sin(z)\backslash exp(iz)$ :$Q(z)=\backslash exp(iz)$ both of which are bounded in the UHP. All of the above examples are of bounded type in the lower half-plane as well, using different ''P'' and ''Q'' functions. But the region mentioned in the definition of the term "bounded type" cannot be the whole complex plane unless the function is constant because one must use the same ''P'' and ''Q'' over the whole region, and the only entire functions (that is, analytic in the whole complex plane) which are bounded are constants, by Liouville's theorem. Another example in the upper half-plane is a "Nevanlinna function", that is, an analytic function that maps the UHP to the closed UHP. If ''f''(''z'') is of this type, then :$f(z)=P(z)/Q(z)$ where ''P'' and ''Q'' are the bounded functions: :$P(z)=\backslash frac$ :$Q(z)=\backslash frac\; 1$ (This obviously applies as well to $f(z)/i$, that is, a function whose real part is non-negative in the UHP.) 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「Bounded type (mathematics)」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.