"자코비 삼중곱(Jacobi triple product)"의 두 판 사이의 차이
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+ | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">이 항목의 스프링노트 원문주소</h5> | ||
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+ | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">개요</h5> | ||
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+ | <math>\sum_{n=-\infty}^\infty z^{n}q^{n^2}= \prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 + zq^{2m-1}\right) \left( 1 + z^{-1}q^{2m-1}\right)</math> | ||
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+ | <math>z=1</math> 인 경우 | ||
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+ | <math>\sum_{n=-\infty}^\infty q^{n^2}= \prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 + q^{2m-1}\right)^2</math> | ||
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+ | (증명) | ||
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+ | [[q-초기하급수(q-hypergeometric series) (통합됨)|q-초기하급수(q-hypergeometric series)]] | ||
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+ | <math>\prod_{n=0}^{\infty}(1+zq^n)=\sum_{n\geq 0}\frac{q^{n(n-1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)} z^n</math> | ||
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+ | <math>\prod_{n=0}^{\infty}\frac{1}{1+zq^n}=\sum_{n\geq 0}\frac{(-1)^n}{(1-q)(1-q^2)\cdots(1-q^n)} z^n</math> | ||
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+ | 를 활용 | ||
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+ | <math>\prod_{m=0}^\infty \left( 1 + zq^{2m+1}\right)=\sum_{n\geq 0}\frac{q^nz^n}{(1-q^2)(1-q^4)\cdots(1-q^{2n})}</math> | ||
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+ | '''[Andrews65] '''참조 ■ | ||
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+ | <h5>재미있는 사실</h5> | ||
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+ | * Math Overflow http://mathoverflow.net/search?q= | ||
+ | * 네이버 지식인 http://kin.search.naver.com/search.naver?where=kin_qna&query= | ||
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+ | <h5>역사</h5> | ||
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+ | * http://www.google.com/search?hl=en&tbs=tl:1&q= | ||
+ | * [http://jeff560.tripod.com/mathword.html Earliest Known Uses of Some of the Words of Mathematics] | ||
+ | * [http://jeff560.tripod.com/mathsym.html Earliest Uses of Various Mathematical Symbols] | ||
+ | * [[수학사연표 (역사)|수학사연표]] | ||
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+ | <h5>메모</h5> | ||
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+ | <h5>관련된 항목들</h5> | ||
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+ | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">수학용어번역</h5> | ||
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+ | * 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]<br> | ||
+ | ** 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://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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+ | <h5>사전 형태의 자료</h5> | ||
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+ | * http://ko.wikipedia.org/wiki/ | ||
+ | * http://en.wikipedia.org/wiki/ | ||
+ | * http://www.proofwiki.org/wiki/ | ||
+ | * http://www.wolframalpha.com/input/?i= | ||
+ | * [http://eom.springer.de/default.htm The Online Encyclopaedia of Mathematics] | ||
+ | * [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions] | ||
+ | * [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences] | ||
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+ | <h5>관련논문</h5> | ||
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+ | * '''[Andrews65]'''[http://www.jstor.org/stable/2033875 Shorter Notes: A Simple Proof of Jacobi's Triple Product Identity]<br> | ||
+ | ** George E. Andrews, Proceedings of the American Mathematical Society, Vol. 16, No. 2 (Apr., 1965), pp. 333-334 | ||
+ | * [http://www.jstor.org/stable/2320552 An Easy Proof of the Triple-Product Identity]<br> | ||
+ | ** John A. Ewell, <cite style="line-height: 2em;">[http://www.jstor.org/action/showPublication?journalCode=amermathmont The American Mathematical Monthly]</cite>, Vol. 88, No. 4 (Apr., 1981), pp. 270-272 | ||
+ | * http://www.jstor.org/action/doBasicSearch?Query= | ||
+ | * http://www.ams.org/mathscinet | ||
+ | * http://dx.doi.org/ | ||
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+ | <h5>관련도서</h5> | ||
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+ | * 도서내검색<br> | ||
+ | ** http://books.google.com/books?q= | ||
+ | ** http://book.daum.net/search/contentSearch.do?query= | ||
+ | * 도서검색<br> | ||
+ | ** http://books.google.com/books?q= | ||
+ | ** http://book.daum.net/search/mainSearch.do?query= | ||
+ | ** http://book.daum.net/search/mainSearch.do?query= | ||
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+ | <h5>관련기사</h5> | ||
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+ | * 네이버 뉴스 검색 (키워드 수정)<br> | ||
+ | ** http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query= | ||
+ | ** http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query= | ||
+ | ** http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query= | ||
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+ | <h5>링크</h5> | ||
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+ | * 구글 블로그 검색<br> | ||
+ | ** http://blogsearch.google.com/blogsearch?q= | ||
+ | * [http://navercast.naver.com/science/list 네이버 오늘의과학] | ||
+ | * [http://www.ams.org/mathmoments/ Mathematical Moments from the AMS] | ||
+ | * [http://betterexplained.com/ BetterExplained] | ||
+ | * [http://www.exampleproblems.com/ exampleproblems.com] |
2010년 12월 29일 (수) 10:05 판
이 항목의 스프링노트 원문주소
개요
\(\sum_{n=-\infty}^\infty z^{n}q^{n^2}= \prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 + zq^{2m-1}\right) \left( 1 + z^{-1}q^{2m-1}\right)\)
\(z=1\) 인 경우
\(\sum_{n=-\infty}^\infty q^{n^2}= \prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 + q^{2m-1}\right)^2\)
(증명)
q-초기하급수(q-hypergeometric series)
\(\prod_{n=0}^{\infty}(1+zq^n)=\sum_{n\geq 0}\frac{q^{n(n-1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)} z^n\)
\(\prod_{n=0}^{\infty}\frac{1}{1+zq^n}=\sum_{n\geq 0}\frac{(-1)^n}{(1-q)(1-q^2)\cdots(1-q^n)} z^n\)
를 활용
\(\prod_{m=0}^\infty \left( 1 + zq^{2m+1}\right)=\sum_{n\geq 0}\frac{q^nz^n}{(1-q^2)(1-q^4)\cdots(1-q^{2n})}\)
[Andrews65] 참조 ■
재미있는 사실
- Math Overflow http://mathoverflow.net/search?q=
- 네이버 지식인 http://kin.search.naver.com/search.naver?where=kin_qna&query=
역사
- http://www.google.com/search?hl=en&tbs=tl:1&q=
- Earliest Known Uses of Some of the Words of Mathematics
- Earliest Uses of Various Mathematical Symbols
- 수학사연표
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/
- http://www.proofwiki.org/wiki/
- http://www.wolframalpha.com/input/?i=
- The Online Encyclopaedia of Mathematics
- NIST Digital Library of Mathematical Functions
- The On-Line Encyclopedia of Integer Sequences
관련논문
- [Andrews65]Shorter Notes: A Simple Proof of Jacobi's Triple Product Identity
- George E. Andrews, Proceedings of the American Mathematical Society, Vol. 16, No. 2 (Apr., 1965), pp. 333-334
- An Easy Proof of the Triple-Product Identity
- John A. Ewell, The American Mathematical Monthly, Vol. 88, No. 4 (Apr., 1981), pp. 270-272
- http://www.jstor.org/action/doBasicSearch?Query=
- http://www.ams.org/mathscinet
- http://dx.doi.org/
관련도서
- 도서내검색
- 도서검색
관련기사
- 네이버 뉴스 검색 (키워드 수정)