"셀베르그 적분(Selberg integral)"의 두 판 사이의 차이
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* 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] | ||
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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]. | * 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]. | ||
2014년 9월 29일 (월) 19:17 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
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
- 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.
- 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.