|
Schröder's equation,〔 ASIN: B0006BTAC2〕 named after Ernst Schröder, is a functional equation with one independent variable: given the function , find the function such that: Schröder's equation is an eigenvalue equation for the composition operator , which sends a function to . If is a fixed point of , meaning , then either (or ) or =1. Thus, provided is finite and does not vanish or diverge, the eigenvalue is given by . ==Functional significance== For , if is analytic on the unit disk, fixes 0, and 0 < || < 1, then Koenigs showed in 1884 that there is an analytic (non-trivial) satisfying Schröder's equation. This is one of the first steps in a long line of theorems fruitful for understanding composition operators on analytic function spaces, cf. Koenigs function. Equations such as Schröder's are suitable to encoding self-similarity, and have thus been extensively utilized in studies of nonlinear dynamics (often referred to colloquially as ''chaos theory''). It is also used in studies of turbulence, as well as the renormalization group. An equivalent transpose form of Schröder's equation for the inverse of Schröder's conjugacy function is . The change of variables (the Abel function) further converts Schröder's equation to the older Abel equation, . Similarly, the change of variables converts Schröder's equation to Böttcher's equation, . Moreover, for the velocity,〔 , ''Julia's equation'', , holds. The ''n''-th power of a solution of Schröder's equation provides a solution of Schröder's equation with eigenvalue , instead. In the same vein, for an invertible solution of Schröder's equation, the (non-invertible) function is also a solution, for ''any'' periodic function with period . All solutions of Schröder's equation are related in this manner. 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「Schröder's equation」の詳細全文を読む スポンサード リンク
|