감마곱 (Gamma Products)
이 항목의 수학노트 원문주소
개요
- 자연수 n에 대한 잉여계의 부분집합 A에 대하여, 다음과 같은 감마함수의 곱이 언제 닫힌 형태로 표현되는가의 문제
\(\prod_{k\in A}\Gamma(\frac{k}{n})\)
예
\(\Gamma \left(\frac{1}{6}\right) \Gamma \left(\frac{5}{6}\right)=2\sqrt{\pi }\)
\(\Gamma \left(\frac{1}{10}\right) \Gamma \left(\frac{3}{10}\right) \Gamma \left(\frac{7}{10}\right) \Gamma \left(\frac{9}{10}\right)=4 \pi ^2\)
\(\Gamma \left(\frac{1}{14}\right) \Gamma \left(\frac{9}{14}\right) \Gamma \left(\frac{11}{14}\right)=4{\pi ^{3/2}}\)
\(\Gamma \left(\frac{3}{14}\right) \Gamma \left(\frac{5}{14}\right) \Gamma \left(\frac{13}{14}\right)=2\pi ^{3/2}\)
\(\Gamma \left(\frac{1}{18}\right) \Gamma \left(\frac{5}{18}\right) \Gamma \left(\frac{7}{18}\right) \Gamma \left(\frac{11}{18}\right) \Gamma \left(\frac{13}{18}\right) \Gamma \left(\frac{17}{18}\right)=8 \pi ^3\)
\(\Gamma \left(\frac{1}{22}\right) \Gamma \left(\frac{3}{22}\right) \Gamma \left(\frac{5}{22}\right) \Gamma \left(\frac{7}{22}\right) \Gamma \left(\frac{9}{22}\right) \Gamma \left(\frac{13}{22}\right) \Gamma \left(\frac{15}{22}\right) \Gamma \left(\frac{17}{22}\right) \Gamma \left(\frac{19}{22}\right) \Gamma \left(\frac{21}{22}\right)=32 \pi ^5\)
\(\Gamma \left(\frac{1}{26}\right) \Gamma \left(\frac{3}{26}\right) \Gamma \left(\frac{5}{26}\right) \Gamma \left(\frac{7}{26}\right) \Gamma \left(\frac{9}{26}\right) \Gamma \left(\frac{11}{26}\right) \Gamma \left(\frac{15}{26}\right) \Gamma \left(\frac{17}{26}\right) \Gamma \left(\frac{19}{26}\right) \Gamma \left(\frac{21}{26}\right) \Gamma \left(\frac{23}{26}\right) \Gamma \left(\frac{25}{26}\right)=64 \pi ^6\)
\(\Gamma \left(\frac{1}{30}\right) \Gamma \left(\frac{17}{30}\right) \Gamma \left(\frac{19}{30}\right) \Gamma \left(\frac{23}{30}\right)=8 \pi ^2\)
\(\Gamma \left(\frac{1}{34}\right) \Gamma \left(\frac{9}{34}\right) \Gamma \left(\frac{13}{34}\right) \Gamma \left(\frac{15}{34}\right) \Gamma \left(\frac{19}{34}\right) \Gamma \left(\frac{21}{34}\right) \Gamma \left(\frac{25}{34}\right) \Gamma \left(\frac{33}{34}\right)=16 \pi ^4\)
재미있는 사실
- Math Overflow http://mathoverflow.net/search?q=
- 네이버 지식인 http://kin.search.naver.com/search.naver?where=kin_qna&query=
역사
메모
관련된 항목들
수학용어번역
- 단어사전 http://www.google.com/dictionary?langpair=en%7Cko&q=
- 발음사전 http://www.forvo.com/search/
- 대한수학회 수학 학술 용어집
- 남·북한수학용어비교
- 대한수학회 수학용어한글화 게시판
사전 형태의 자료
- http://ko.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/
- The Online Encyclopaedia of Mathematics
- NIST Digital Library of Mathematical Functions
- The World of Mathematical Equations
관련논문
- Luschny, Peter, and Stefan Wehmeier. 2009. “The lcm(1,2,...,n) as a product of sine values sampled over the points in Farey sequences.” 0909.1838 (September 10). http://arxiv.org/abs/0909.1838 .
- Martin, Greg. 2009. “A product of Gamma function values at fractions with the same denominator.” 0907.4384 (July 24). http://arxiv.org/abs/0907.4384 .
- Nijenhuis, Albert. 2009. “Small Gamma Products with Simple Values.” 0907.1689 (July 9). http://arxiv.org/abs/0907.1689 .
- Problem 11426, M. L. Glasser, The American Mathematical Monthly, Vol. 116, No. 4 (Apr., 2009), p. 365 http://www.jstor.org/stable/40391099
- http://www.jstor.org/action/doBasicSearch?Query=
- http://www.ams.org/mathscinet
- http://dx.doi.org/
관련도서