"맥도날드-메타 적분"의 두 판 사이의 차이
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| + | ==개요== | ||
| + | * <math>G</math>는 <math>\mathbb{R}^n</math>에 작용하는 유한반사군 (콕세터군) | ||
| + | * <math>R</math>은 루트시스템, <math>R_+</math>는 양의 루트 | ||
| + | * <math>(\cdot,\cdot)</math>는 <math>\mathbb{R}^n</math>에 정의된 <math>(\alpha,\alpha)=2,\,\alpha\in R</math>을 만족하는 내적 | ||
| + | * 다음이 성립한다 | ||
| + | :<math> | ||
| + | \int_{\Bbb R''}\prod_{\alpha \in R_+} |(\alpha,x)|^{2 k}\, d\varphi(x)=\prod_{j=1}^n\frac{\Gamma(1+k d_j)}{\Gamma(1+k)}, | ||
| + | </math> | ||
| + | 여기서 <math>d_i</math>는 [[콕세터 군의 차수와 지수 (degrees and exponents)|콕세터군의 차수]]이고 <math>\varphi</math>는 <math>\mathbb{R}^n</math>에 정의된 가우스 측도 | ||
| + | :<math> | ||
| + | d\varphi(x):=\frac{e^{-|x|^2/2}}{(2\pi)^{n/2}}\, | ||
| + | d x_1\cdots d x_n | ||
| + | </math> | ||
| + | |||
| + | |||
| + | ==관련된 항목들== | ||
| + | * [[셀베르그 적분(Selberg integral)]] | ||
| + | |||
| + | |||
| + | ==매스매티카 파일 및 계산리소스== | ||
| + | * https://docs.google.com/file/d/0B8XXo8Tve1cxSi1QSGF0bGdTaU0/edit | ||
| + | * http://mathpro.blogspot.com.au/2008/01/selberg-integral.html | ||
| + | |||
==리뷰, 에세이, 강의노트== | ==리뷰, 에세이, 강의노트== | ||
* P. Etingof, [http://ocw.mit.edu/courses/mathematics/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/lecture-notes/MIT18_735F09_ch04.pdf The Macdonald-Mehta integral] | * P. Etingof, [http://ocw.mit.edu/courses/mathematics/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/lecture-notes/MIT18_735F09_ch04.pdf The Macdonald-Mehta integral] | ||
* S. Ole Warnaar, [http://www.maths.adelaide.edu.au/thomas.leistner/colloquium/20110805OleWarnaar/Selberg.pdf The Selberg Integral], 2011 | * S. Ole Warnaar, [http://www.maths.adelaide.edu.au/thomas.leistner/colloquium/20110805OleWarnaar/Selberg.pdf The Selberg Integral], 2011 | ||
* S. Ole Warnaar, [http://www.maths.uq.edu.au/%7Euqowarna/talks/Wien.pdf Beta Integrals] | * S. Ole Warnaar, [http://www.maths.uq.edu.au/%7Euqowarna/talks/Wien.pdf Beta Integrals] | ||
| + | |||
| + | |||
| + | ==관련논문== | ||
| + | * Brent, Richard P., Christian Krattenthaler, and S. Ole Warnaar. “Discrete Analogues of Macdonald-Mehta Integrals.” arXiv:1601.06536 [math-Ph], January 25, 2016. http://arxiv.org/abs/1601.06536. | ||
| + | * Cacciatori, S. L., F. Dalla Piazza, and A. Scotti. “Compact Lie Groups: Euler Constructions and Generalized Dyson Conjecture.” arXiv:1207.1262 [hep-Th, Physics:math-Ph], July 5, 2012. http://arxiv.org/abs/1207.1262. | ||
| + | * Brent, Richard P., Hideyuki Ohtsuka, Judy-anne H. Osborn, and Helmut Prodinger. “Some Binomial Sums Involving Absolute Values.” arXiv:1411.1477 [math, Stat], November 5, 2014. http://arxiv.org/abs/1411.1477. | ||
| + | * Opdam, E. M. “Dunkl Operators, Bessel Functions and the Discriminant of a Finite Coxeter Group.” Compositio Mathematica 85, no. 3 (1993): 333–73. | ||
| + | * Opdam, E. M. “Some Applications of Hypergeometric Shift Operators.” Inventiones Mathematicae 98, no. 1 (February 1989): 1–18. doi:10.1007/BF01388841. | ||
| + | * Garvan, Frank G. “Some Macdonald-Mehta Integrals by Brute Force.” In <math>q</math>-Series and Partitions (Minneapolis, MN, 1988), 18:77–98. IMA Vol. Math. Appl. Springer, New York, 1989. http://www.ams.org/mathscinet-getitem?mr=1019845. | ||
| + | [[분류:적분]] | ||
2020년 11월 16일 (월) 05:21 기준 최신판
개요
- \(G\)는 \(\mathbb{R}^n\)에 작용하는 유한반사군 (콕세터군)
- \(R\)은 루트시스템, \(R_+\)는 양의 루트
- \((\cdot,\cdot)\)는 \(\mathbb{R}^n\)에 정의된 \((\alpha,\alpha)=2,\,\alpha\in R\)을 만족하는 내적
- 다음이 성립한다
\[ \int_{\Bbb R''}\prod_{\alpha \in R_+} |(\alpha,x)|^{2 k}\, d\varphi(x)=\prod_{j=1}^n\frac{\Gamma(1+k d_j)}{\Gamma(1+k)}, \] 여기서 \(d_i\)는 콕세터군의 차수이고 \(\varphi\)는 \(\mathbb{R}^n\)에 정의된 가우스 측도 \[ d\varphi(x):=\frac{e^{-|x|^2/2}}{(2\pi)^{n/2}}\, d x_1\cdots d x_n \]
관련된 항목들
매스매티카 파일 및 계산리소스
- https://docs.google.com/file/d/0B8XXo8Tve1cxSi1QSGF0bGdTaU0/edit
- http://mathpro.blogspot.com.au/2008/01/selberg-integral.html
리뷰, 에세이, 강의노트
- P. Etingof, The Macdonald-Mehta integral
- S. Ole Warnaar, The Selberg Integral, 2011
- S. Ole Warnaar, Beta Integrals
관련논문
- Brent, Richard P., Christian Krattenthaler, and S. Ole Warnaar. “Discrete Analogues of Macdonald-Mehta Integrals.” arXiv:1601.06536 [math-Ph], January 25, 2016. http://arxiv.org/abs/1601.06536.
- Cacciatori, S. L., F. Dalla Piazza, and A. Scotti. “Compact Lie Groups: Euler Constructions and Generalized Dyson Conjecture.” arXiv:1207.1262 [hep-Th, Physics:math-Ph], July 5, 2012. http://arxiv.org/abs/1207.1262.
- Brent, Richard P., Hideyuki Ohtsuka, Judy-anne H. Osborn, and Helmut Prodinger. “Some Binomial Sums Involving Absolute Values.” arXiv:1411.1477 [math, Stat], November 5, 2014. http://arxiv.org/abs/1411.1477.
- Opdam, E. M. “Dunkl Operators, Bessel Functions and the Discriminant of a Finite Coxeter Group.” Compositio Mathematica 85, no. 3 (1993): 333–73.
- Opdam, E. M. “Some Applications of Hypergeometric Shift Operators.” Inventiones Mathematicae 98, no. 1 (February 1989): 1–18. doi:10.1007/BF01388841.
- Garvan, Frank G. “Some Macdonald-Mehta Integrals by Brute Force.” In \(q\)-Series and Partitions (Minneapolis, MN, 1988), 18:77–98. IMA Vol. Math. Appl. Springer, New York, 1989. http://www.ams.org/mathscinet-getitem?mr=1019845.