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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 Poisson–Jensen formula ： ウィキペディア英語版 Jensen's formula In the mathematical field known as complex analysis, Jensen's formula, introduced by , relates the average magnitude of an analytic function on a circle with the number of its zeros inside the circle. It forms an important statement in the study of entire functions. == The statement == Suppose that ''ƒ'' is an analytic function in a region in the complex plane which contains the closed disk D of radius ''r'' about the origin, ''a''1, ''a''2, ..., ''a''''n'' are the zeros of ''ƒ'' in the interior of D repeated according to multiplicity, and ''ƒ''(0) ≠ 0. Jensen's formula states that :$\backslash log\; |f(0)|\; =\; \backslash sum\_^n\; \backslash log\; \backslash left(\; \backslash frac\backslash right)\; +\; \backslash frac\; \backslash int\_0^\; \backslash log|f(re^)|\; \backslash ,\; d\backslash theta.$ This formula establishes a connection between the moduli of the zeros of the function ''ƒ'' inside the disk D and the average of ''log |f(z)|'' on the boundary circle |''z''| = ''r'', and can be seen as a generalisation of the mean value property of harmonic functions. Namely, if ''f'' has no zeros in D, then Jensen's formula reduces to :$\backslash log\; |f(0)|\; =\; \backslash frac\; \backslash int\_0^\; \backslash log|f(re^)|\; \backslash ,\; d\backslash theta,$ which is the mean-value property of the harmonic function $\backslash log\; |f(z)|$. An equivalent statement of Jensen's formula that is frequently used is :$\backslash frac\; \backslash int\_0^\; \backslash log\; |f(re^)|\; \backslash ;\; d\backslash theta$ - \log |f(0)| = \int_0^r \frac \; dt where $n(t)$ denotes the number of zeros of $f$ in the disc of radius $t$ centered at the origin. Jensen's formula may be generalized for functions which are merely meromorphic on D. Namely, assume that :$f(z)=z^l\; \backslash frac,$ where ''g'' and ''h'' are analytic functions in D having zeros at $a\_1,\backslash ldots,a\_n\; \backslash in\; \backslash mathbb\; D\backslash backslash\backslash$ and $b\_1,\backslash ldots,b\_m\; \backslash in\; \backslash mathbb\; D\backslash backslash\backslash$ respectively, then Jensen's formula for meromorphic functions states that :$\backslash log\; \backslash left|\backslash frac\backslash right|\; =\; \backslash log\; \backslash left\; |r^\; \backslash frac\backslash right|\; +\; \backslash frac\; \backslash int\_0^\; \backslash log|f(re^)|\; \backslash ,\; d\backslash theta.$ Jensen's formula can be used to estimate the number of zeros of analytic function in a circle. Namely, if ''f'' is a function analytic in a disk of radius ''R'' centered at ''z0'' and if ''|f|'' is bounded by ''M'' on the boundary of that disk, then the number of zeros of ''f'' in a circle of radius ''r''<''R'' centered at the same point ''z0'' does not exceed :$$ \frac \log \frac. Jensen's formula is an important statement in the study of value distribution of entire and meromorphic functions. In particular, it is the starting point of Nevanlinna theory. 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「Jensen's formula」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.