English Dictionary

analytic function

名詞

(mathematical analysis) Any smooth (infinitely differentiable) function f, defined on an open set D⊆ℂ( textit or⊆ℝ), whose value in some neighbourhood of any given point x_0∈D is given by the Taylor series ∑ₙ₌₀ ᪲(f⁽ⁿ⁾(x_0))/(n!)(x-x_0)ⁿ.

日本語の意味

解析関数。任意の点の近傍において、その点でのテイラー級数展開（無限項の級数）によって表現できる、無限回微分可能な関数。数学的解析、特に複素解析や実解析で用いられる概念。

An analytic function defined on an open subset of the complex plane is completely determined by its Taylor series around any point in that domain.

複素平面の開集合上で定義された解析関数は、その領域内の任意の点のまわりのテイラー級数によって完全に決定される。

意味（1）

(mathematical analysis) Any smooth (infinitely differentiable) function f, defined on an open set D⊆ℂ( textit or⊆ℝ), whose value in some neighbourhood of any given point x_0∈D is given by the Taylor series ∑ₙ₌₀ ᪲(f⁽ⁿ⁾(x_0))/(n!)(x-x_0)ⁿ.

There is a large and important literature concerned with the question of how to characterize precisely the boundary values of analytic functions. For example, when the analytic functions are of Hardy Class H* in a half plane, then their boundary values are attained in the L2 norm and, in fact, the analytic function can be reproduced by a Cauchy integral of the boundary value.

analytic functions （ plural ）