"나스랄라-라만 적분"의 두 판 사이의 차이
Pythagoras0 (토론 | 기여) (→관련논문) |
Pythagoras0 (토론 | 기여) |
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| 1번째 줄: | 1번째 줄: | ||
==개요== | ==개요== | ||
| − | + | ;정리 (나스랄라-라만 Nassrallah-Rahman) | |
| − | + | 복소수 $t_1, \dots ,t_5,q$가 $|t_1|, \dots , |t_5|,|q| <1$을 만족한다고 하자. 다음이 성립한다. | |
\begin{equation}\label{NR} | \begin{equation}\label{NR} | ||
\frac{(q,q)_\infty}{2} \int_{\mathbb{T}}\frac{(z \prod_{i=1}^5 t_i,q)_\infty (z^{-1} \prod_{i=1}^5 t_i,q)_\infty (z^2,q)_\infty (z^{-2},q)_\infty}{\prod_{i=1}^5 (t_i z)_\infty (t_i z^{-1})_\infty} \frac{dz}{2\pi i z} \ = \ \frac{\prod_{j=1}^5 (\frac{t_1 t_2 t_3 t_4 t_5}{t_j},q)_\infty}{\prod_{1 \leq i < j \leq 5} (t_i t_j,q)_\infty} | \frac{(q,q)_\infty}{2} \int_{\mathbb{T}}\frac{(z \prod_{i=1}^5 t_i,q)_\infty (z^{-1} \prod_{i=1}^5 t_i,q)_\infty (z^2,q)_\infty (z^{-2},q)_\infty}{\prod_{i=1}^5 (t_i z)_\infty (t_i z^{-1})_\infty} \frac{dz}{2\pi i z} \ = \ \frac{\prod_{j=1}^5 (\frac{t_1 t_2 t_3 t_4 t_5}{t_j},q)_\infty}{\prod_{1 \leq i < j \leq 5} (t_i t_j,q)_\infty} | ||
\end{equation} | \end{equation} | ||
| + | * 나스랄라-라만 삼각 베타 적분 (Nassrallah-Rahman trigonometric beta integral)으로 불린다 | ||
| + | * $t_1\to 0$일 때, 애스키-윌슨 적분을 얻는다 | ||
==확장== | ==확장== | ||
;정리 (Spiridonov). | ;정리 (Spiridonov). | ||
| − | + | 복소수 $t_1, \dots ,t_6,p,q$가 $|t_1|, \dots , |t_6|,|p|,|q| <1$이고, $\prod_{i=1}^6 t_i=pq$을 만족한다고 하자. 다음이 성립한다. | |
\begin{equation} \label{betaint} | \begin{equation} \label{betaint} | ||
\frac{(p;p)_\infty (q;q)_\infty}{2} \int_{\mathbb{T}} \frac{\prod_{i=1}^6 \Gamma(t_i z ;p,q)\Gamma(t_i z^{-1} ;p,q)}{\Gamma(z^{2};p,q) \Gamma(z^{-2};p,q)} \frac{dz}{2 \pi i z} = \prod_{1 \leq i < j \leq 6} \Gamma(t_i t_j;p,q), | \frac{(p;p)_\infty (q;q)_\infty}{2} \int_{\mathbb{T}} \frac{\prod_{i=1}^6 \Gamma(t_i z ;p,q)\Gamma(t_i z^{-1} ;p,q)}{\Gamma(z^{2};p,q) \Gamma(z^{-2};p,q)} \frac{dz}{2 \pi i z} = \prod_{1 \leq i < j \leq 6} \Gamma(t_i t_j;p,q), | ||
\end{equation} | \end{equation} | ||
| − | where the unit circle $\mathbb{T}$ is taken in the positive orientation | + | where the unit circle $\mathbb{T}$ is taken in the positive orientation |
* $p \rightarrow 0$일 때, \ref{NR}을 얻는다 | * $p \rightarrow 0$일 때, \ref{NR}을 얻는다 | ||
| 20번째 줄: | 22번째 줄: | ||
* observed by Rahman in \cite{Rahman2} as a special case of the integral found in \cite{Nasrallah} | * observed by Rahman in \cite{Rahman2} as a special case of the integral found in \cite{Nasrallah} | ||
| + | |||
| + | ==리뷰, 에세이, 강의노트== | ||
| + | * Gahramanov, Ilmar. “Mathematical Structures behind Supersymmetric Dualities.” arXiv:1505.05656 [hep-Th, Physics:math-Ph], May 21, 2015. http://arxiv.org/abs/1505.05656. | ||
2015년 8월 10일 (월) 20:18 판
개요
- 정리 (나스랄라-라만 Nassrallah-Rahman)
복소수 $t_1, \dots ,t_5,q$가 $|t_1|, \dots , |t_5|,|q| <1$을 만족한다고 하자. 다음이 성립한다. \begin{equation}\label{NR} \frac{(q,q)_\infty}{2} \int_{\mathbb{T}}\frac{(z \prod_{i=1}^5 t_i,q)_\infty (z^{-1} \prod_{i=1}^5 t_i,q)_\infty (z^2,q)_\infty (z^{-2},q)_\infty}{\prod_{i=1}^5 (t_i z)_\infty (t_i z^{-1})_\infty} \frac{dz}{2\pi i z} \ = \ \frac{\prod_{j=1}^5 (\frac{t_1 t_2 t_3 t_4 t_5}{t_j},q)_\infty}{\prod_{1 \leq i < j \leq 5} (t_i t_j,q)_\infty} \end{equation}
- 나스랄라-라만 삼각 베타 적분 (Nassrallah-Rahman trigonometric beta integral)으로 불린다
- $t_1\to 0$일 때, 애스키-윌슨 적분을 얻는다
확장
- 정리 (Spiridonov).
복소수 $t_1, \dots ,t_6,p,q$가 $|t_1|, \dots , |t_6|,|p|,|q| <1$이고, $\prod_{i=1}^6 t_i=pq$을 만족한다고 하자. 다음이 성립한다. \begin{equation} \label{betaint} \frac{(p;p)_\infty (q;q)_\infty}{2} \int_{\mathbb{T}} \frac{\prod_{i=1}^6 \Gamma(t_i z ;p,q)\Gamma(t_i z^{-1} ;p,q)}{\Gamma(z^{2};p,q) \Gamma(z^{-2};p,q)} \frac{dz}{2 \pi i z} = \prod_{1 \leq i < j \leq 6} \Gamma(t_i t_j;p,q), \end{equation} where the unit circle $\mathbb{T}$ is taken in the positive orientation
- $p \rightarrow 0$일 때, \ref{NR}을 얻는다
메모
- observed by Rahman in \cite{Rahman2} as a special case of the integral found in \cite{Nasrallah}
리뷰, 에세이, 강의노트
- Gahramanov, Ilmar. “Mathematical Structures behind Supersymmetric Dualities.” arXiv:1505.05656 [hep-Th, Physics:math-Ph], May 21, 2015. http://arxiv.org/abs/1505.05656.
관련논문
- Nassrallah, B., and M. Rahman. “Projection Formulas, a Reproducing Kernel and a Generating Function for Q-Wilson Polynomials.” SIAM Journal on Mathematical Analysis 16, no. 1 (January 1, 1985): 186–97. doi:10.1137/0516014.
- Askey, Richard, and James Arthur Wilson. Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Vol. 319. American Mathematical Soc., 1985.