"에어리 (Airy) 함수와 미분방정식"의 두 판 사이의 차이

2021년 2월 17일 (수) 04:53 기준 최신판

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=

수학용어번역

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

매스매티카 파일 및 계산 리소스

사전 형태의 자료

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.

메타데이터

위키데이터

ID : Q409415

Spacy 패턴 목록

[{'LOWER': 'airy'}, {'LEMMA': 'function'}]
[{'LOWER': 'airy'}, {'LEMMA': 'function'}]
[{'LOWER': 'airy'}, {'LOWER': 'differential'}, {'LEMMA': 'equation'}]
[{'LOWER': 'airy'}, {'LEMMA': 'integral'}]
[{'LOWER': 'airy'}, {'LOWER': 'function'}, {'LOWER': 'of'}, {'LOWER': 'the'}, {'LOWER': 'first'}, {'LEMMA': 'kind'}]

@@ 1번째 줄: / 1번째 줄: @@
 ==개요==
-* <math>y'' - xy = 0</math>
+* 에어리 미분방정식<math>y'' - xy = 0</math>
+* 에어리 함수 <math>Ai,Bi</math>는 일차독립인 두 해이다
+:<math>\mathrm{Ai}(x) = \frac{1}{\pi} \int_0^\infty \cos\left(\tfrac13t^3 + xt\right)\, dt,</math>
+:<math>\mathrm{Bi}(x) = \frac{1}{\pi} \int_0^\infty \left[\exp\left(-\tfrac13t^3 + xt\right) + \sin\left(\tfrac13t^3 + xt\right)\,\right]dt.,</math>
-<math>\mathrm{Ai}(x) = \frac{1}{\pi} \int_0^\infty \cos\left(\tfrac13t^3 + xt\right)\, dt,</math>
-<math>\mathrm{Bi}(x) = \frac{1}{\pi} \int_0^\infty \left[\exp\left(-\tfrac13t^3 + xt\right) + \sin\left(\tfrac13t^3 + xt\right)\,\right]dt.,</math>
-http://www.wolframalpha.com/input/?i=Ai%28x%29
 ==근사공식==
+* [[안장점 근사]]
-* [[안장점 근사]]<br><math>x>>0</math> 일 때,<br><math>\mathrm{Ai}(x) \sim \frac{e^{-\frac{2 x^{3/2}}{3}}}{2 \sqrt{\pi } \sqrt[4]{x}}</math><br><math>x<<0</math> 일 때,<br><math>\mathrm{Ai}(x) \sim  \frac{\sin \left(\frac{2 |x|^{3/2}}{3}+\frac{\pi }{4}\right)}{\sqrt{\pi } \sqrt[4]{|x|}}</math><br>
+* <math>x>>0</math> 일 때,:<math>\mathrm{Ai}(x) \sim \frac{e^{-\frac{2 x^{3/2}}{3}}}{2 \sqrt{\pi }x^{1/4}}</math>
-* [http://www.math.umn.edu/%7Eymori/docs/teaching/fall08/airy.pdf Asymptotics of the Airy Function]<br>
+* <math>x<<0</math> 일 때,:<math>\mathrm{Ai}(x) \sim  \frac{\sin \left(\frac{2 |x|^{3/2}}{3}+\frac{\pi }{4}\right)}{\sqrt{\pi } \sqrt[4]{|x|}}</math>
+* [http://www.math.umn.edu/%7Eymori/docs/teaching/fall08/airy.pdf Asymptotics of the Airy Function]
 ==역사==
@@ 33번째 줄: / 18번째 줄: @@
 * http://www.google.com/search?hl=en&tbs=tl:1&q=
-* [[수학사연표 (역사)|수학사연표]]
+* [[수학사 연표]]
@@ 44번째 줄: / 29번째 줄: @@
 * Math Overflow http://mathoverflow.net/search?q=
 ==관련된 항목들==
+* 점근 급수(asymptotic series)
 ==수학용어번역==
-*  단어사전<br>
-** http://www.google.com/dictionary?langpair=en|ko&q=
 ** http://ko.wiktionary.org/wiki/
-*  발음사전<br>
+*  발음사전
 ** http://www.forvo.com/word/airy/#en
 ** 아이어리?
-* [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집]<br>
-** http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
-* [http://www.nktech.net/science/term/term_l.jsp?l_mode=cate&s_code_cd=MA 남·북한수학용어비교]
-* [http://kms.or.kr/home/kor/board/bulletin_list_subject.asp?bulletinid=%7BD6048897-56F9-43D7-8BB6-50B362D1243A%7D&boardname=%BC%F6%C7%D0%BF%EB%BE%EE%C5%E4%B7%D0%B9%E6&globalmenu=7&localmenu=4 대한수학회 수학용어한글화 게시판]
@@ 75번째 줄: / 47번째 줄: @@
 * https://docs.google.com/file/d/0B8XXo8Tve1cxbl96STk2T3dpajg/edit
-* http://www.wolframalpha.com/input/?i=
+* http://www.wolframalpha.com/input/?i=Ai%28x%29
-* http://functions.wolfram.com/
-* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
-* [http://people.math.sfu.ca/%7Ecbm/aands/toc.htm Abramowitz and Stegun Handbook of mathematical functions]
-* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
-* [http://numbers.computation.free.fr/Constants/constants.html Numbers, constants and computation]
-* [https://docs.google.com/open?id=0B8XXo8Tve1cxMWI0NzNjYWUtNmIwZi00YzhkLTkzNzQtMDMwYmVmYmIxNmIw 매스매티카 파일 목록]
@@ 101번째 줄: / 66번째 줄: @@
 ==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.
-==관련논문==
+[[분류:미분방정식]]
-* http://www.jstor.org/action/doBasicSearch?Query=
+==메타데이터==
-* http://www.ams.org/mathscinet
+===위키데이터===
-* http://dx.doi.org/
+* ID :  [https://www.wikidata.org/wiki/Q409415 Q409415]
+===Spacy 패턴 목록===
+* [{'LOWER': 'airy'}, {'LEMMA': 'function'}]
+* [{'LOWER': 'airy'}, {'LEMMA': 'function'}]
+* [{'LOWER': 'airy'}, {'LOWER': 'differential'}, {'LEMMA': 'equation'}]
+* [{'LOWER': 'airy'}, {'LEMMA': 'integral'}]
+* [{'LOWER': 'airy'}, {'LOWER': 'function'}, {'LOWER': 'of'}, {'LOWER': 'the'}, {'LOWER': 'first'}, {'LEMMA': 'kind'}]