"셀베르그 적분(Selberg integral)"의 두 판 사이의 차이
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==관련논문== | ==관련논문== | ||
| + | * 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. | * Rains, Eric M. “Multivariate Quadratic Transformations and the Interpolation Kernel.” arXiv:1408.0305 [math], August 1, 2014. http://arxiv.org/abs/1408.0305. | ||
* 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. “The $\mathfrak{sl}_3$ Selberg Integral.” Advances in Mathematics 224, no. 2 (2010): 499–524. doi:10.1016/j.aim.2009.11.011. | ||
2014년 12월 3일 (수) 06:24 판
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
\[S_{1} (\alpha, \beta,\gamma)=B(\alpha,\beta) = \int_0^1t^{\alpha-1}(1-t)^{\beta-1}\,dt\]
역사
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
사전 형태의 자료
리뷰, 에세이, 강의노트
- S. Ole Warnaar, The Selberg Integral, 2011
- S. Ole 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
- 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.
- 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.
- 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.
- Selberg, Atle. “Remarks on a Multiple Integral.” Norsk Mat. Tidsskr. 26 (1944): 71–78.