"모듈라 군, j-invariant and the singular moduli"의 두 판 사이의 차이
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<math>\lambda(i)=k^2(i)=\frac{1}{2}</math> | <math>\lambda(i)=k^2(i)=\frac{1}{2}</math> | ||
− | [[ | + | [[제1종타원적분 K (complete elliptic integral of the first kind)|일종타원적분 K]] |
<math>K(k) = \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-k^2 \sin^2\theta}}</math> | <math>K(k) = \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-k^2 \sin^2\theta}}</math> | ||
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− | + | * [[타원적분|타원적분, 타원함수, 타원곡선]] | |
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+ | * [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=&fstr= ][http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=kor_term&fstr=%EB%AA%A8%EB%93%88%EB%9D%BC http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=kor_term&fstr=모듈라]<br> | ||
+ | ** http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=modular<br> | ||
+ | ** [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=modular ]http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=<br> | ||
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<h5>참고할만한 자료</h5> | <h5>참고할만한 자료</h5> | ||
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* [http://www.math.lsu.edu/%7Everrill/fundomain/ Fundamental Domain drawer]<br> | * [http://www.math.lsu.edu/%7Everrill/fundomain/ Fundamental Domain drawer]<br> | ||
** Java applet | ** Java applet | ||
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** Roger C. Alperin, <cite>The American Mathematical Monthly</cite>, Vol. 106, No. 8 (Oct., 1999), pp. 771-773 | ** Roger C. Alperin, <cite>The American Mathematical Monthly</cite>, Vol. 106, No. 8 (Oct., 1999), pp. 771-773 | ||
* [http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=262662 On singular moduli.]<br> | * [http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=262662 On singular moduli.]<br> | ||
+ | ** <br> | ||
** Gross, B.H.; Zagier, Don B, J. Rcinc Angew. Math. 355, 191-220 | ** Gross, B.H.; Zagier, Don B, J. Rcinc Angew. Math. 355, 191-220 | ||
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2009년 11월 17일 (화) 11:05 판
간단한 소개
\(k=k(\tau)=\frac{\theta_2^2(\tau)}{\theta_3^2(\tau)}\)
\(k'=\sqrt{1-k^2}=\frac{\theta_4^2(\tau)}{\theta_3^2(\tau)}\)
\(\lambda(\tau)=k^2(\tau)=\frac{\theta_2^4(\tau)}{\theta_3^4(\tau)}\)
\(J(\tau)=\frac{4}{27}\frac{(1-\lambda+\lambda^2)^3}{\lambda^2(1-\lambda)^2}\)
\(j(\tau)=1728J(\tau)\)
- \(\lambda(\tau)=k^2(\tau)\) 는 modulus라고 불렸으며, 아벨, 자코비와 후학들(에르미트)에 의해 많이 연구됨
- \(\Gamma(2)\)에 의해 불변임
- 기본적인 내용은 [AHL1979] 7.3.4를 참고
- 가장 기본적인 모듈라함수로 여겨졌으나, 나중에 \(j\)-불변량에 그 자리를 내줌
메모
\(\lambda(i)=k^2(i)=\frac{1}{2}\)
\(K(k) = \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-k^2 \sin^2\theta}}\)
하위페이지
관련된 항목들
수학용어번역
표준적인 도서 및 추천도서
- Discontinuous Groups and Automorphic Functions
- Joseph Lehner
- [AHL1979]Complex Analysis
- Lars Ahlfors, 3rd edition, McGraw-Hill, 1979
위키링크
참고할만한 자료
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- Fundamental Domain drawer
- Java applet
- H. A. Verrill
- The Action of the Modular Group on the Fundamental Domain
- Wolfram
- Modular Miracles
- John Stillwell, The American Mathematical Monthly, Vol. 108, No. 1 (Jan., 2001), pp. 70-76
- Rationals and the Modular Group
- Roger C. Alperin, The American Mathematical Monthly, Vol. 106, No. 8 (Oct., 1999), pp. 771-773
- On singular moduli.
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- Gross, B.H.; Zagier, Don B, J. Rcinc Angew. Math. 355, 191-220
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