에어리 (Airy) 함수와 미분방정식

개요

• 에어리 미분방정식\(y'' - xy = 0\)
• 에어리 함수 $Ai,Bi$는 일차독립인 두 해이다

\[\mathrm{Ai}(x) = \frac{1}{\pi} \int_0^\infty \cos\left(\tfrac13t^3 + xt\right)\, dt,\] \[\mathrm{Bi}(x) = \frac{1}{\pi} \int_0^\infty \left[\exp\left(-\tfrac13t^3 + xt\right) + \sin\left(\tfrac13t^3 + xt\right)\,\right]dt.,\]

근사공식

• 안장점 근사
• \(x>>0\) 일 때,\[\mathrm{Ai}(x) \sim \frac{e^{-\frac{2 x^{3/2}}{3}}}{2 \sqrt{\pi }x^{1/4}}\]
• \(x<<0\) 일 때,\[\mathrm{Ai}(x) \sim \frac{\sin \left(\frac{2 |x|^{3/2}}{3}+\frac{\pi }{4}\right)}{\sqrt{\pi } \sqrt[4]{|x|}}\]
  
• Asymptotics of the Airy Function
  

역사

• http://www.google.com/search?hl=en&tbs=tl:1&q=
• 수학사 연표

메모

• Math Overflow http://mathoverflow.net/search?q=

관련된 항목들

• 점근 급수(asymptotic series)

수학용어번역

• • http://ko.wiktionary.org/wiki/
• 발음사전
  
  • http://www.forvo.com/word/airy/#en
  • 아이어리?

매스매티카 파일 및 계산 리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxbl96STk2T3dpajg/edit
• http://www.wolframalpha.com/input/?i=Ai%28x%29

사전 형태의 자료

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Airy_equation
• http://en.wikipedia.org/wiki/WKB_approximation
• The Online Encyclopaedia of Mathematics
• NIST Digital Library of Mathematical Functions
• The World of Mathematical Equations

리뷰논문, 에세이, 강의노트

• http://www.ams.org/samplings/feature-column/fcarc-rainbows
• Duistermaat, J. J. “The Light in the Neighborhood of a Caustic.” In Séminaire Bourbaki Vol. 1976/77 Exposés 489–506, 19–29. Lecture Notes in Mathematics 677. Springer Berlin Heidelberg, 1978. http://link.springer.com/chapter/10.1007/BFb0070750.

관련논문

• Clarkson, Peter A. “On Airy Solutions of the Second Painlev’e Equation.” arXiv:1510.08326 [nlin], October 28, 2015. http://arxiv.org/abs/1510.08326.
• Duistermaat, J. J. “Oscillatory Integrals, Lagrange Immersions and Unfolding of Singularities.” Communications on Pure and Applied Mathematics 27 (1974): 207–81.
• On the Intensity of Light in the Neighborhood of a Caustic, 1838. http://archive.org/details/cbarchive_36815_ontheintensityoflightintheneig1838.