셀베르그 적분(Selberg integral)

개요

• 오일러 베타적분의 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

 

• http://www.google.com/search?hl=en&tbs=tl:1&q=
• 수학사 연표
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관련된 항목들

• 오일러 베타적분
  
• Chowla-셀베르그 공식
  

 

 

사전 형태의 자료

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Selberg_integral

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