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
메모
- Math Overflow http://mathoverflow.net/search?q=
관련된 항목들
- 점근 급수(asymptotic series)
수학용어번역
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.
메타데이터
위키데이터
- ID : Q409415