"셀베르그 적분(Selberg integral)"의 두 판 사이의 차이
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==개요== | ==개요== | ||
| − | * [[오일러 베타적분(베타함수) | + | * [[오일러 베타적분(베타함수)]]의 일반화 |
:<math> | :<math> | ||
\begin{align} S_{n} (\alpha, \beta, \gamma) & = | \begin{align} S_{n} (\alpha, \beta, \gamma) & = | ||
| 13번째 줄: | 13번째 줄: | ||
:<math>S_{1} (\alpha, \beta,\gamma)=B(\alpha,\beta) = \int_0^1t^{\alpha-1}(1-t)^{\beta-1}\,dt</math> | :<math>S_{1} (\alpha, \beta,\gamma)=B(\alpha,\beta) = \int_0^1t^{\alpha-1}(1-t)^{\beta-1}\,dt</math> | ||
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==역사== | ==역사== | ||
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* http://www.google.com/search?hl=en&tbs=tl:1&q= | * http://www.google.com/search?hl=en&tbs=tl:1&q= | ||
* [[수학사 연표]] | * [[수학사 연표]] | ||
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==메모== | ==메모== | ||
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==관련된 항목들== | ==관련된 항목들== | ||
| − | * [[오일러 베타적분(베타함수)|오일러 베타적분]] | + | * [[오일러 베타적분(베타함수)|오일러 베타적분]] |
| − | * [[Chowla-셀베르그 공식]] | + | * [[맥도날드-메타 적분]] |
| + | * [[Chowla-셀베르그 공식]] | ||
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| − | + | ==사전 형태의 자료== | |
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| − | ==사전 | ||
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* http://en.wikipedia.org/wiki/Selberg_integral | * http://en.wikipedia.org/wiki/Selberg_integral | ||
| 55번째 줄: | 49번째 줄: | ||
* 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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==관련논문== | ==관련논문== | ||
* 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. | ||
| − | * [http://dx.doi.org/10.1007/s11005-009-0330-7 On a Selberg–Schur Integral] | + | * [http://dx.doi.org/10.1007/s11005-009-0330-7 On a Selberg–Schur Integral] |
** Sergio Manuel Iguri, 2009 | ** Sergio Manuel Iguri, 2009 | ||
| − | * [http://www.ams.org/journals/bull/2008-45-04/S0273-0979-08-01221-4/home.html The importance of the Selberg integral] | + | * [http://www.ams.org/journals/bull/2008-45-04/S0273-0979-08-01221-4/home.html The importance of the Selberg integral] |
** Peter J. Forrester; S. Ole Warnaar, Bull. Amer. Math. Soc. 45 (2008), 489-534. | ** Peter J. Forrester; S. Ole Warnaar, Bull. Amer. Math. Soc. 45 (2008), 489-534. | ||
| − | * [http://dx.doi.org/10.1088/0305-4470/36/19/306 Hankel hyperdeterminants and Selberg integrals] | + | * [http://dx.doi.org/10.1088/0305-4470/36/19/306 Hankel hyperdeterminants and Selberg integrals] |
** J.-G. Luque, J.-Y. Thibon, 2002 | ** J.-G. Luque, J.-Y. Thibon, 2002 | ||
2014년 9월 26일 (금) 02:58 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
관련논문
- Rains, Eric M. “Multivariate Quadratic Transformations and the Interpolation Kernel.” arXiv:1408.0305 [math], August 1, 2014. http://arxiv.org/abs/1408.0305.
- On a Selberg–Schur Integral
- Sergio Manuel Iguri, 2009
- The importance of the Selberg integral
- Peter J. Forrester; S. Ole Warnaar, Bull. Amer. Math. Soc. 45 (2008), 489-534.
- Hankel hyperdeterminants and Selberg integrals
- J.-G. Luque, J.-Y. Thibon, 2002