"셀베르그 적분(Selberg integral)"의 두 판 사이의 차이

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==개요==
  
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* [[오일러 베타적분(베타함수)]]의 일반화
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:<math>
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\begin{align} S_{n} (\alpha, \beta, \gamma) & =
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\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 \\
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& = \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>
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여기서
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:<math>
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\Re(\alpha)>0, \Re(\beta)>0, \Re(\gamma)>\max\{-\frac{1}{n},-\frac{\Re{\alpha}}{n-1},-\frac{\Re{\beta}}{n-1}\}
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</math>
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*  n=1 인 경우
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:<math>S_{1} (\alpha, \beta,\gamma)=B(\alpha,\beta) = \int_0^1t^{\alpha-1}(1-t)^{\beta-1}\,dt</math>
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==메모==
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* Algebra (Coxeter groups, double affine Hecke algebras)
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* Conformal field theory (KZ equations)
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* Gauge theory (supersymmetry, AGT conjecture)
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* Geometry (hyperplane arrangements)
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* Number theory (moments <math>\zeta(s)</math>
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* Orthogonal polynomials (Generalised Jacobi polynomials)
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* Random matrices
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* Statistics
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* Statistical physics
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==관련된 항목들==
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* [[오일러 베타적분(베타함수)|오일러 베타적분]]
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* [[맥도날드-메타 적분]]
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* [[타원 셀베르그 적분]]
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* [[Chowla-셀베르그 공식]]
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==매스매티카 파일 및 계산 리소스==
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* https://drive.google.com/file/d/0B8XXo8Tve1cxLVdyVDk2N0Yydjg/view
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==사전 형태의 자료==
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* http://en.wikipedia.org/wiki/Selberg_integral
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==리뷰, 에세이, 강의노트==
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* 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
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* Warnaar, [http://www.maths.adelaide.edu.au/thomas.leistner/colloquium/20110805OleWarnaar/Selberg.pdf The Selberg Integral], 2011
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* Warnaar, [http://www.maths.uq.edu.au/~uqowarna/talks/FPSAC08.pdf The Mukhin{Varchenko conjecture for type A], 2008
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* Warnaar, [http://www.maths.uq.edu.au/%7Euqowarna/talks/Wien.pdf Beta Integrals]
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* 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].
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==관련논문==
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* 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
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* 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.
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* Patterson, Samuel J. “Selberg Sums - a New Perspective.” arXiv:1411.7600 [math], November 27, 2014. http://arxiv.org/abs/1411.7600.
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* Rains, Eric M. “Multivariate Quadratic Transformations and the Interpolation Kernel.” arXiv:1408.0305 [math], August 1, 2014. http://arxiv.org/abs/1408.0305.
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* 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.
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* 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.
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* 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.
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* 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.
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* 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.
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* 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.
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* 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.
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* 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.
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* 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.
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* Selberg, Atle. “Remarks on a Multiple Integral.” Norsk Mat. Tidsskr. 26 (1944): 71–78.
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[[분류:적분]]
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[[분류:특수함수]]
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==메타데이터==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q7447525 Q7447525]
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===Spacy 패턴 목록===
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* [{'LOWER': 'selberg'}, {'LEMMA': 'integral'}]

2021년 2월 17일 (수) 04:48 기준 최신판

개요

\[ \begin{align} S_{n} (\alpha, \beta, \gamma) & = \int_0^1 \cdots \int_0^1 \prod_{i=1}^n t_i^{\alpha-1}(1-t_i)^{\beta-1} \prod_{1 \le i < j \le n} |t_i - t_j |^{2 \gamma}\,dt_1 \cdots dt_n \\ & = \prod_{j = 0}^{n-1} \frac {\Gamma(\alpha + j \gamma) \Gamma(\beta + j \gamma) \Gamma (1 + (j+1)\gamma)} {\Gamma(\alpha + \beta + (n+j-1)\gamma) \Gamma(1+\gamma)} \end{align},\] 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

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

Spacy 패턴 목록

  • [{'LOWER': 'selberg'}, {'LEMMA': 'integral'}]
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