"바이어슈트라스 타원함수 ℘"의 두 판 사이의 차이
27번째 줄: | 27번째 줄: | ||
<math>=\frac{1}{z}+\sum_{\omega \in \Omega}(-\frac{z^2}{\omega^3}-\frac{z^3}{\omega^4}-\frac{z^4}{\omega^5}-\cdots)=\frac{1}{z}-G_3z^2-G_4z^3-\cdots=\frac{1}{z}-\sum_{n=2}^{\infty}G_{2n}z^{2n-1}</math>. 여기서 <math>G_{2n}=\sum_{\omega\in \Omega} \frac{1}{\omega^{2n}}</math>. | <math>=\frac{1}{z}+\sum_{\omega \in \Omega}(-\frac{z^2}{\omega^3}-\frac{z^3}{\omega^4}-\frac{z^4}{\omega^5}-\cdots)=\frac{1}{z}-G_3z^2-G_4z^3-\cdots=\frac{1}{z}-\sum_{n=2}^{\infty}G_{2n}z^{2n-1}</math>. 여기서 <math>G_{2n}=\sum_{\omega\in \Omega} \frac{1}{\omega^{2n}}</math>. | ||
− | 따라서 <math>\wp(z)= | + | 따라서 <math>\wp(z)=\frac{1}{z^2}-\sum_{n=2}^{\infty}(2n-1)G_{2n}z^{2n-2}</math>. |
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+ | * <math>G_{2n}</math>에 대해서는 <br> | ||
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+ | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">간단한 소개</h5> | ||
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+ | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">상위 주제</h5> | ||
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+ | ==== 하위페이지 ==== | ||
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+ | * [[1964250|0 토픽용템플릿]]<br> | ||
+ | ** [[2060652|0 상위주제템플릿]]<br> | ||
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+ | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">재미있는 사실</h5> | ||
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+ | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">역사</h5> | ||
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+ | * [[수학사연표 (역사)|수학사연표]]<br> | ||
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+ | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">많이 나오는 질문과 답변</h5> | ||
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+ | * 네이버 지식인<br> | ||
+ | ** http://kin.search.naver.com/search.naver?where=kin_qna&query= | ||
+ | ** http://kin.search.naver.com/search.naver?where=kin_qna&query= | ||
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+ | ** http://kin.search.naver.com/search.naver?where=kin_qna&query= | ||
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+ | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">관련된 고교수학 또는 대학수학</h5> | ||
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+ | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">관련된 다른 주제들</h5> | ||
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+ | <br> | ||
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+ | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">관련도서 및 추천도서</h5> | ||
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+ | * 도서내검색<br> | ||
+ | ** http://books.google.com/books?q= | ||
+ | ** http://book.daum.net/search/contentSearch.do?query= | ||
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+ | * http://ko.wikipedia.org/wiki/ | ||
+ | * http://en.wikipedia.org/wiki/ | ||
+ | * http://www.wolframalpha.com/input/?i= | ||
+ | * http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q= | ||
+ | * http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7= | ||
+ | * 다음백과사전 http://enc.daum.net/dic100/search.do?q= | ||
+ | * [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집] | ||
+ | * [http://navercast.naver.com/science/list 네이버 오늘의과학] | ||
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+ | * 네이버 뉴스 검색 (키워드 수정)<br> | ||
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+ | * 구글 블로그 검색 http://blogsearch.google.com/blogsearch?q= | ||
+ | * 트렌비 블로그 검색 http://www.trenb.com/search.qst?q= | ||
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+ | |||
+ | * http://commons.wikimedia.org/w/index.php?title=Special%3ASearch&search= | ||
+ | * http://images.google.com/images?q= | ||
+ | * [http://www.artchive.com/ http://www.artchive.com] | ||
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+ | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">동영상</h5> | ||
+ | |||
+ | * http://www.youtube.com/results?search_type=&search_query= | ||
+ | * <br> |
2009년 7월 2일 (목) 21:48 판
정의
- 2차원격자를 이루는 두 복소수 \(\omega_1,\omega_2\)에 대하여,
\(\wp(z;\omega_1,\omega_2)=\frac{1}{z^2}+ \sum_{m^2+n^2 \ne 0} \left\{ \frac{1}{(z-m\omega_1-n\omega_2)^2}- \frac{1}{\left(m\omega_1+n\omega_2\right)^2} \right\}\)
\(\wp(z;\omega_1,\omega_2)=\frac{1}{z^2}+ \sum_{m^2+n^2 \ne 0} \left\{ \frac{1}{(z-m\omega_1-n\omega_2)^2}- \frac{1}{\left(m\omega_1+n\omega_2\right)^2} \right\}\)
- 는 타원함수가 됨.
\(\wp\)의 로랑급수
- 원점에서의 로랑급수는 다음과 같이 주어짐.
\(\wp(z)=z^{-2}+\frac{1}{20}g_2z^2+\frac{1}{28}g_3z^4+O(z^6)\)
여기서 \(g_2= 60\sum{}' \Omega_{m,n}^{-4},\qquad g_3=140\sum{}' \Omega_{m,n}^{-6}\)
(증명)
\(\zeta(z)=\frac{1}{z}+\sum_{\omega \in \Omega}\frac{z^2}{\omega^2(z-\omega)}=\frac{1}{z}+\sum_{\omega \in \Omega}(\frac{1}{z-\omega}+\frac{1}{\omega}+\frac{z}{\omega}^2) \) 를 정의하자.
\(\wp(z)=-\zeta'(z)=\sum_{\omega\in \Omega} \frac{1}{(z-m)^2}- \frac{1}{\omega^2} \right\}\) 이므로 \(\zeta(z)=\frac{1}{z}+\sum_{\Omega}(\frac{1}{z-\omega}+\frac{1}{\omega}+\frac{z}{\omega}^2)\) 의 로랑급수를 구한 뒤, 미분을 하면 된다.
\(\zeta(z)=\frac{1}{z}+\sum_{\omega \in \Omega}\frac{z^2}{\omega^2(z-\omega)}=\frac{1}{z}+\sum_{\omega \in \Omega}\frac{z^2}{\omega^2}(-\frac{1}{\omega}-\frac{z}{\omega^2}-\frac{z^2}{\omega^3}-\cdots)\)
\(=\frac{1}{z}+\sum_{\omega \in \Omega}(-\frac{z^2}{\omega^3}-\frac{z^3}{\omega^4}-\frac{z^4}{\omega^5}-\cdots)=\frac{1}{z}-G_3z^2-G_4z^3-\cdots=\frac{1}{z}-\sum_{n=2}^{\infty}G_{2n}z^{2n-1}\). 여기서 \(G_{2n}=\sum_{\omega\in \Omega} \frac{1}{\omega^{2n}}\).
따라서 \(\wp(z)=\frac{1}{z^2}-\sum_{n=2}^{\infty}(2n-1)G_{2n}z^{2n-2}\).
- \(G_{2n}\)에 대해서는
간단한 소개
상위 주제
하위페이지
재미있는 사실
역사
많이 나오는 질문과 답변
- 네이버 지식인
- http://kin.search.naver.com/search.naver?where=kin_qna&query=
- http://kin.search.naver.com/search.naver?where=kin_qna&query=
- http://kin.search.naver.com/search.naver?where=kin_qna&query=
- http://kin.search.naver.com/search.naver?where=kin_qna&query=
- http://kin.search.naver.com/search.naver?where=kin_qna&query=
관련된 고교수학 또는 대학수학
관련된 다른 주제들
관련도서 및 추천도서
- 도서내검색
- 도서검색
참고할만한 자료
- http://ko.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/
- http://www.wolframalpha.com/input/?i=
- http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
- http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
- 다음백과사전 http://enc.daum.net/dic100/search.do?q=
- 대한수학회 수학 학술 용어집
- 네이버 오늘의과학
관련기사
- 네이버 뉴스 검색 (키워드 수정)
- 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=
- 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=
블로그
- 구글 블로그 검색 http://blogsearch.google.com/blogsearch?q=
- 트렌비 블로그 검색 http://www.trenb.com/search.qst?q=
이미지 검색
- http://commons.wikimedia.org/w/index.php?title=Special%3ASearch&search=
- http://images.google.com/images?q=
- http://www.artchive.com