"감마곱 (Gamma Products)"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) |
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| (사용자 2명의 중간 판 12개는 보이지 않습니다) | |||
| 1번째 줄: | 1번째 줄: | ||
| − | + | ==개요== | |
| − | * | + | * 자연수 n에 대한 잉여계의 부분집합 A에 대하여, 다음과 같은 감마함수의 곱이 언제 닫힌 형태로 표현되는가의 문제:<math>\prod_{k\in A}\Gamma(\frac{k}{n})</math> |
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| − | + | ==예== | |
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<math>\Gamma \left(\frac{1}{6}\right) \Gamma \left(\frac{5}{6}\right)=2\sqrt{\pi }</math> | <math>\Gamma \left(\frac{1}{6}\right) \Gamma \left(\frac{5}{6}\right)=2\sqrt{\pi }</math> | ||
| − | + | <math>\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</math> | |
<math>\Gamma \left(\frac{1}{14}\right) \Gamma \left(\frac{9}{14}\right) \Gamma \left(\frac{11}{14}\right)=4{\pi ^{3/2}}</math> | <math>\Gamma \left(\frac{1}{14}\right) \Gamma \left(\frac{9}{14}\right) \Gamma \left(\frac{11}{14}\right)=4{\pi ^{3/2}}</math> | ||
| 23번째 줄: | 17번째 줄: | ||
<math>\Gamma \left(\frac{3}{14}\right) \Gamma \left(\frac{5}{14}\right) \Gamma \left(\frac{13}{14}\right)=2\pi ^{3/2}</math> | <math>\Gamma \left(\frac{3}{14}\right) \Gamma \left(\frac{5}{14}\right) \Gamma \left(\frac{13}{14}\right)=2\pi ^{3/2}</math> | ||
| − | + | <math>\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</math> | |
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| − | + | <math>\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</math> | |
| − | < | + | <math>\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</math> |
| − | + | <math>\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</math> | |
| − | + | <math>\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</math> | |
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| + | ==메모== | ||
| − | + | * http://mathoverflow.net/questions/9878/a-product-of-gamma-values-over-the-numbers-coprime-to-n | |
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| − | + | ==관련된 항목들== | |
| − | + | * [[Chowla-셀베르그 공식]] | |
| + | * [[1부터 n까지의 최소공배수]] | ||
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| − | + | ==수학용어번역== | |
| − | * | + | * 단어사전 http://www.google.com/dictionary?langpair=en|ko&q= |
| − | * | + | * 발음사전 http://www.forvo.com/search/ |
| − | * [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집] | + | * [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집] |
** http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr= | ** http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr= | ||
* [http://www.nktech.net/science/term/term_l.jsp?l_mode=cate&s_code_cd=MA 남·북한수학용어비교] | * [http://www.nktech.net/science/term/term_l.jsp?l_mode=cate&s_code_cd=MA 남·북한수학용어비교] | ||
| − | * [http://kms.or.kr/home/kor/board/bulletin_list_subject.asp?bulletinid=%7BD6048897-56F9-43D7-8BB6-50B362D1243A%7D&boardname=%BC%F6%C7%D0%BF%EB%BE%EE%C5%E4%B7%D0%B9%E6&globalmenu=7&localmenu=4 | + | * [http://kms.or.kr/home/kor/board/bulletin_list_subject.asp?bulletinid=%7BD6048897-56F9-43D7-8BB6-50B362D1243A%7D&boardname=%BC%F6%C7%D0%BF%EB%BE%EE%C5%E4%B7%D0%B9%E6&globalmenu=7&localmenu=4 대한수학회 수학용어한글화 게시판] |
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| − | + | ==사전 형태의 자료== | |
* http://ko.wikipedia.org/wiki/ | * http://ko.wikipedia.org/wiki/ | ||
| 84번째 줄: | 68번째 줄: | ||
* [http://eqworld.ipmnet.ru/ The World of Mathematical Equations] | * [http://eqworld.ipmnet.ru/ The World of Mathematical Equations] | ||
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| − | + | ==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.” <em>0909.1838</em> (September 10). http://arxiv.org/abs/0909.1838 . | * Luschny, Peter, and Stefan Wehmeier. 2009. “The lcm(1,2,...,n) as a product of sine values sampled over the points in Farey sequences.” <em>0909.1838</em> (September 10). http://arxiv.org/abs/0909.1838 . | ||
| − | * Martin, Greg. 2009. “A product of Gamma function values at fractions with the same denominator.” <em>0907.4384</em> (July 24). http://arxiv.org/abs/0907.4384 . | + | * Martin, Greg. 2009. “A product of Gamma function values at fractions with the same denominator.” <em>0907.4384</em> (July 24). http://arxiv.org/abs/0907.4384 . |
* Nijenhuis, Albert. 2009. “Small Gamma Products with Simple Values.” <em>0907.1689</em> (July 9). http://arxiv.org/abs/0907.1689 . | * Nijenhuis, Albert. 2009. “Small Gamma Products with Simple Values.” <em>0907.1689</em> (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 | + | * Problem 11426, M. L. Glasser, The American Mathematical Monthly, Vol. 116, No. 4 (Apr., 2009), p. 365 http://www.jstor.org/stable/40391099 |
| + | ** solution [http://math.la.asu.edu/%7Echeckman/AMM/11426_Heckman.pdf http://math.la.asu.edu/~checkman/AMM/11426_Heckman.pdf] | ||
* http://www.jstor.org/action/doBasicSearch?Query= | * http://www.jstor.org/action/doBasicSearch?Query= | ||
* http://www.ams.org/mathscinet | * http://www.ams.org/mathscinet | ||
* http://dx.doi.org/ | * http://dx.doi.org/ | ||
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2020년 12월 28일 (월) 02:03 기준 최신판
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
\(\Gamma \left(\frac{1}{6}\right) \Gamma \left(\frac{5}{6}\right)=2\sqrt{\pi }\)
\(\Gamma \left(\frac{1}{10}\right) \Gamma \left(\frac{3}{10}\right) \Gamma \left(\frac{7}{10}\right) \Gamma \left(\frac{9}{10}\right)=4 \pi ^2\)
\(\Gamma \left(\frac{1}{14}\right) \Gamma \left(\frac{9}{14}\right) \Gamma \left(\frac{11}{14}\right)=4{\pi ^{3/2}}\)
\(\Gamma \left(\frac{3}{14}\right) \Gamma \left(\frac{5}{14}\right) \Gamma \left(\frac{13}{14}\right)=2\pi ^{3/2}\)
\(\Gamma \left(\frac{1}{18}\right) \Gamma \left(\frac{5}{18}\right) \Gamma \left(\frac{7}{18}\right) \Gamma \left(\frac{11}{18}\right) \Gamma \left(\frac{13}{18}\right) \Gamma \left(\frac{17}{18}\right)=8 \pi ^3\)
\(\Gamma \left(\frac{1}{22}\right) \Gamma \left(\frac{3}{22}\right) \Gamma \left(\frac{5}{22}\right) \Gamma \left(\frac{7}{22}\right) \Gamma \left(\frac{9}{22}\right) \Gamma \left(\frac{13}{22}\right) \Gamma \left(\frac{15}{22}\right) \Gamma \left(\frac{17}{22}\right) \Gamma \left(\frac{19}{22}\right) \Gamma \left(\frac{21}{22}\right)=32 \pi ^5\)
\(\Gamma \left(\frac{1}{26}\right) \Gamma \left(\frac{3}{26}\right) \Gamma \left(\frac{5}{26}\right) \Gamma \left(\frac{7}{26}\right) \Gamma \left(\frac{9}{26}\right) \Gamma \left(\frac{11}{26}\right) \Gamma \left(\frac{15}{26}\right) \Gamma \left(\frac{17}{26}\right) \Gamma \left(\frac{19}{26}\right) \Gamma \left(\frac{21}{26}\right) \Gamma \left(\frac{23}{26}\right) \Gamma \left(\frac{25}{26}\right)=64 \pi ^6\)
\(\Gamma \left(\frac{1}{30}\right) \Gamma \left(\frac{17}{30}\right) \Gamma \left(\frac{19}{30}\right) \Gamma \left(\frac{23}{30}\right)=8 \pi ^2\)
\(\Gamma \left(\frac{1}{34}\right) \Gamma \left(\frac{9}{34}\right) \Gamma \left(\frac{13}{34}\right) \Gamma \left(\frac{15}{34}\right) \Gamma \left(\frac{19}{34}\right) \Gamma \left(\frac{21}{34}\right) \Gamma \left(\frac{25}{34}\right) \Gamma \left(\frac{33}{34}\right)=16 \pi ^4\)
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/