"셀베르그 적분(Selberg integral)"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) |
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| (사용자 2명의 중간 판 39개는 보이지 않습니다) | |||
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| − | + | ==개요== | |
| − | * [[ | + | * [[오일러 베타적분(베타함수)]]의 일반화 |
| + | :<math> | ||
| + | \begin{align} S_{n} (\alpha, \beta, \gamma) & = | ||
| + | \int_0^1 \cdots \int_0^1 \prod_{i=1}^n t_i^{\alpha-1}(1-t_i)^{\beta-1} \prod_{1 \le i < j \le n} |t_i - t_j |^{2 \gamma}\,dt_1 \cdots dt_n \\ | ||
| + | & = \prod_{j = 0}^{n-1} \frac {\Gamma(\alpha + j \gamma) \Gamma(\beta + j \gamma) \Gamma (1 + (j+1)\gamma)} {\Gamma(\alpha + \beta + (n+j-1)\gamma) \Gamma(1+\gamma)} \end{align},</math> | ||
| + | 여기서 | ||
| + | :<math> | ||
| + | \Re(\alpha)>0, \Re(\beta)>0, \Re(\gamma)>\max\{-\frac{1}{n},-\frac{\Re{\alpha}}{n-1},-\frac{\Re{\beta}}{n-1}\} | ||
| + | </math> | ||
| + | * n=1 인 경우 | ||
| + | :<math>S_{1} (\alpha, \beta,\gamma)=B(\alpha,\beta) = \int_0^1t^{\alpha-1}(1-t)^{\beta-1}\,dt</math> | ||
| − | |||
| − | + | ==메모== | |
| + | * Algebra (Coxeter groups, double affine Hecke algebras) | ||
| + | * Conformal field theory (KZ equations) | ||
| + | * Gauge theory (supersymmetry, AGT conjecture) | ||
| + | * Geometry (hyperplane arrangements) | ||
| + | * Number theory (moments <math>\zeta(s)</math> | ||
| + | * Orthogonal polynomials (Generalised Jacobi polynomials) | ||
| + | * Random matrices | ||
| + | * Statistics | ||
| + | * Statistical physics | ||
| − | + | ==관련된 항목들== | |
| + | * [[오일러 베타적분(베타함수)|오일러 베타적분]] | ||
| + | * [[맥도날드-메타 적분]] | ||
| + | * [[타원 셀베르그 적분]] | ||
| + | * [[Chowla-셀베르그 공식]] | ||
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| − | + | ==매스매티카 파일 및 계산 리소스== | |
| + | * https://drive.google.com/file/d/0B8XXo8Tve1cxLVdyVDk2N0Yydjg/view | ||
| − | + | ==사전 형태의 자료== | |
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* http://en.wikipedia.org/wiki/Selberg_integral | * http://en.wikipedia.org/wiki/Selberg_integral | ||
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| − | + | ==리뷰, 에세이, 강의노트== | |
| + | * Alessandro Zaccagnini, The Selberg integral and a new pair-correlation function for the zeros of the Riemann zeta-function, http://arxiv.org/abs/1603.02952v1 | ||
| + | * Warnaar, [http://www.maths.adelaide.edu.au/thomas.leistner/colloquium/20110805OleWarnaar/Selberg.pdf The Selberg Integral], 2011 | ||
| + | * Warnaar, [http://www.maths.uq.edu.au/~uqowarna/talks/FPSAC08.pdf The Mukhin{Varchenko conjecture for type A], 2008 | ||
| + | * Warnaar, [http://www.maths.uq.edu.au/%7Euqowarna/talks/Wien.pdf Beta Integrals] | ||
| + | * Forrester, Peter, and S. Warnaar. “The Importance of the Selberg Integral.” Bulletin of the American Mathematical Society 45, no. 4 (2008): 489–534. doi:[http://www.ams.org/journals/bull/2008-45-04/S0273-0979-08-01221-4/home.html 10.1090/S0273-0979-08-01221-4]. | ||
| − | + | ==관련논문== | |
| + | * Peter J. Forrester, Volumes for <math>{\rm SL}_N(\mathbb R)</math>, the Selberg integral and random lattices, arXiv:1604.07462 [math-ph], April 25 2016, http://arxiv.org/abs/1604.07462 | ||
| + | * Rosengren, Hjalmar. “Selberg Integrals, Askey-Wilson Polynomials and Lozenge Tilings of a Hexagon with a Triangular Hole.” arXiv:1503.00971 [math], March 3, 2015. http://arxiv.org/abs/1503.00971. | ||
| + | * Patterson, Samuel J. “Selberg Sums - a New Perspective.” arXiv:1411.7600 [math], November 27, 2014. http://arxiv.org/abs/1411.7600. | ||
| + | * Rains, Eric M. “Multivariate Quadratic Transformations and the Interpolation Kernel.” arXiv:1408.0305 [math], August 1, 2014. http://arxiv.org/abs/1408.0305. | ||
| + | * Mironov, S., A. Morozov, and Y. Zenkevich. ‘Generalized Jack Polynomials and the AGT Relations for the SU(3) Group’. JETP Letters 99, no. 2 (1 March 2014): 109–13. doi:10.1134/S0021364014020076. | ||
| + | * Zhang, Hong, and Yutaka Matsuo. ‘Selberg Integral and SU(N) AGT Conjecture’. Journal of High Energy Physics 2011, no. 12 (December 2011). doi:10.1007/JHEP12(2011)106. | ||
| + | * Mironov, A., Al Morozov, and And Morozov. ‘Matrix Model Version of AGT Conjecture and Generalized Selberg Integrals’. Nuclear Physics B 843, no. 2 (February 2011): 534–57. doi:10.1016/j.nuclphysb.2010.10.016. | ||
| + | * Warnaar, S. Ole. “The <math>\mathfrak{sl}_3</math> Selberg Integral.” Advances in Mathematics 224, no. 2 (2010): 499–524. doi:10.1016/j.aim.2009.11.011. | ||
| + | * Warnaar, S. Ole. “A Selberg Integral for the Lie Algebra <math>A_n</math>.” Acta Mathematica 203, no. 2 (2009): 269–304. doi:10.1007/s11511-009-0043-x. | ||
| + | * Warnaar, S. Ole. ‘The Mukhin--Varchenko Conjecture for Type A’. DMTCS Proceedings 0, no. 1 (22 December 2008). http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/article/view/dmAJ0108. | ||
| + | * Luque, Jean-Gabriel, and Jean-Yves Thibon. “Hankel Hyperdeterminants and Selberg Integrals.” Journal of Physics A: Mathematical and General 36, no. 19 (May 16, 2003): 5267. doi:10.1088/0305-4470/36/19/306. | ||
| + | * Tarasov, V., and A. Varchenko. ‘Selberg-Type Integrals Associated with SL3’. Letters in Mathematical Physics 65, no. 3 (1 September 2003): 173–85. doi:10.1023/B:MATH.0000010712.67685.9d. | ||
| + | * Gustafson, Robert A. “A Generalization of Selberg’s Beta Integral.” Bulletin (New Series) of the American Mathematical Society 22, no. 1 (January 1990): 97–105. | ||
| + | * Selberg, Atle. “Remarks on a Multiple Integral.” Norsk Mat. Tidsskr. 26 (1944): 71–78. | ||
| − | + | [[분류:적분]] | |
| + | [[분류:특수함수]] | ||
| − | + | ==메타데이터== | |
| − | + | ===위키데이터=== | |
| − | * | + | * ID : [https://www.wikidata.org/wiki/Q7447525 Q7447525] |
| − | + | ===Spacy 패턴 목록=== | |
| − | + | * [{'LOWER': 'selberg'}, {'LEMMA': 'integral'}] | |
| − | * [ | ||
2021년 2월 17일 (수) 05:48 기준 최신판
개요
- 오일러 베타적분(베타함수)의 일반화
\[ \begin{align} S_{n} (\alpha, \beta, \gamma) & = \int_0^1 \cdots \int_0^1 \prod_{i=1}^n t_i^{\alpha-1}(1-t_i)^{\beta-1} \prod_{1 \le i < j \le n} |t_i - t_j |^{2 \gamma}\,dt_1 \cdots dt_n \\ & = \prod_{j = 0}^{n-1} \frac {\Gamma(\alpha + j \gamma) \Gamma(\beta + j \gamma) \Gamma (1 + (j+1)\gamma)} {\Gamma(\alpha + \beta + (n+j-1)\gamma) \Gamma(1+\gamma)} \end{align},\] 여기서 \[ \Re(\alpha)>0, \Re(\beta)>0, \Re(\gamma)>\max\{-\frac{1}{n},-\frac{\Re{\alpha}}{n-1},-\frac{\Re{\beta}}{n-1}\} \]
- n=1 인 경우
\[S_{1} (\alpha, \beta,\gamma)=B(\alpha,\beta) = \int_0^1t^{\alpha-1}(1-t)^{\beta-1}\,dt\]
메모
- Algebra (Coxeter groups, double affine Hecke algebras)
- Conformal field theory (KZ equations)
- Gauge theory (supersymmetry, AGT conjecture)
- Geometry (hyperplane arrangements)
- Number theory (moments \(\zeta(s)\)
- Orthogonal polynomials (Generalised Jacobi polynomials)
- Random matrices
- Statistics
- Statistical physics
관련된 항목들
매스매티카 파일 및 계산 리소스
사전 형태의 자료
리뷰, 에세이, 강의노트
- Alessandro Zaccagnini, The Selberg integral and a new pair-correlation function for the zeros of the Riemann zeta-function, http://arxiv.org/abs/1603.02952v1
- Warnaar, The Selberg Integral, 2011
- Warnaar, The Mukhin{Varchenko conjecture for type A, 2008
- Warnaar, Beta Integrals
- Forrester, Peter, and S. Warnaar. “The Importance of the Selberg Integral.” Bulletin of the American Mathematical Society 45, no. 4 (2008): 489–534. doi:10.1090/S0273-0979-08-01221-4.
관련논문
- Peter J. Forrester, Volumes for \({\rm SL}_N(\mathbb R)\), the Selberg integral and random lattices, arXiv:1604.07462 [math-ph], April 25 2016, http://arxiv.org/abs/1604.07462
- Rosengren, Hjalmar. “Selberg Integrals, Askey-Wilson Polynomials and Lozenge Tilings of a Hexagon with a Triangular Hole.” arXiv:1503.00971 [math], March 3, 2015. http://arxiv.org/abs/1503.00971.
- Patterson, Samuel J. “Selberg Sums - a New Perspective.” arXiv:1411.7600 [math], November 27, 2014. http://arxiv.org/abs/1411.7600.
- Rains, Eric M. “Multivariate Quadratic Transformations and the Interpolation Kernel.” arXiv:1408.0305 [math], August 1, 2014. http://arxiv.org/abs/1408.0305.
- Mironov, S., A. Morozov, and Y. Zenkevich. ‘Generalized Jack Polynomials and the AGT Relations for the SU(3) Group’. JETP Letters 99, no. 2 (1 March 2014): 109–13. doi:10.1134/S0021364014020076.
- Zhang, Hong, and Yutaka Matsuo. ‘Selberg Integral and SU(N) AGT Conjecture’. Journal of High Energy Physics 2011, no. 12 (December 2011). doi:10.1007/JHEP12(2011)106.
- Mironov, A., Al Morozov, and And Morozov. ‘Matrix Model Version of AGT Conjecture and Generalized Selberg Integrals’. Nuclear Physics B 843, no. 2 (February 2011): 534–57. doi:10.1016/j.nuclphysb.2010.10.016.
- Warnaar, S. Ole. “The \(\mathfrak{sl}_3\) Selberg Integral.” Advances in Mathematics 224, no. 2 (2010): 499–524. doi:10.1016/j.aim.2009.11.011.
- Warnaar, S. Ole. “A Selberg Integral for the Lie Algebra \(A_n\).” Acta Mathematica 203, no. 2 (2009): 269–304. doi:10.1007/s11511-009-0043-x.
- Warnaar, S. Ole. ‘The Mukhin--Varchenko Conjecture for Type A’. DMTCS Proceedings 0, no. 1 (22 December 2008). http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/article/view/dmAJ0108.
- Luque, Jean-Gabriel, and Jean-Yves Thibon. “Hankel Hyperdeterminants and Selberg Integrals.” Journal of Physics A: Mathematical and General 36, no. 19 (May 16, 2003): 5267. doi:10.1088/0305-4470/36/19/306.
- Tarasov, V., and A. Varchenko. ‘Selberg-Type Integrals Associated with SL3’. Letters in Mathematical Physics 65, no. 3 (1 September 2003): 173–85. doi:10.1023/B:MATH.0000010712.67685.9d.
- Gustafson, Robert A. “A Generalization of Selberg’s Beta Integral.” Bulletin (New Series) of the American Mathematical Society 22, no. 1 (January 1990): 97–105.
- Selberg, Atle. “Remarks on a Multiple Integral.” Norsk Mat. Tidsskr. 26 (1944): 71–78.
메타데이터
위키데이터
- ID : Q7447525
Spacy 패턴 목록
- [{'LOWER': 'selberg'}, {'LEMMA': 'integral'}]