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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 Airy function ： ウィキペディア英語版 Airy function In the physical sciences, the Airy function Ai(''x'') is a special function named after the British astronomer George Biddell Airy (1801–92). The function Ai(''x'') and the related function Bi(''x''), which is also called the Airy function, but sometimes referred to as the Bairy function, are solutions to the differential equation :$\backslash frac\; -\; xy\; =\; 0\; ,\; \backslash ,\backslash !$ known as the Airy equation or the Stokes equation. This is the simplest second-order linear differential equation with a turning point (a point where the character of the solutions changes from oscillatory to exponential). The Airy function is the solution to Schrödinger's equation for a particle confined within a triangular potential well and for a particle in a one-dimensional constant force field. For the same reason, it also serves to provide uniform semiclassical approximations near a turning point in the WKB approximation, when the potential may be locally approximated by a linear function of position. The triangular potential well solution is directly relevant for the understanding of many semiconductor devices. The Airy function also underlies the form of the intensity near an optical directional caustic, such as that of the rainbow. Historically, this was the mathematical problem that led Airy to develop this special function. The Airy function is also important in microscopy and astronomy; it describes the pattern, due to diffraction and interference, produced by a point source of light (one of which is smaller than the resolution limit of a microscope or telescope). ==Definitions== For real values of ''x'', the Airy function of the first kind can be defined by the improper Riemann integral: :$\backslash mathrm(x)\; =\; \backslash frac\backslash int\_0^\backslash infty\backslash cos\backslash left(\backslash tfrac\; +\; xt\backslash right)\backslash ,\; dt\backslash equiv\; \backslash frac\backslash lim\_\; \backslash int\_0^b\; \backslash cos\backslash left(\backslash tfrac\; +\; xt\backslash right)\backslash ,\; dt,$ which converges because the positive and negative parts of the rapid oscillations tend to cancel one another out (as can be checked by integration by parts). ''y'' = Ai(''x'') satisfies the Airy equation :$y\text{'}\text{'}\; -\; xy\; =\; 0.$ This equation has two linearly independent solutions. Up to scalar multiplication, Ai(''x'') is the solution subject to the condition ''y'' → 0 as ''x'' → ∞. The standard choice for the other solution is the Airy function of the second kind, denoted Bi(''x''). It is defined as the solution with the same amplitude of oscillation as Ai(''x'') as ''x'' → −∞ which differs in phase by π/2: :$\backslash mathrm(x)\; =\; \backslash frac\; \backslash int\_0^\backslash infty\; \backslash left(+\; xt\backslash right)\; +\; \backslash sin\backslash left(\backslash tfrac\; +\; xt\backslash right)\backslash ,\backslash right)dt.$ 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「Airy 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.