"베버(Weber) 모듈라 함수"의 두 판 사이의 차이

수학노트
둘러보기로 가기 검색하러 가기
1번째 줄: 1번째 줄:
−
<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>
+
==개요==
−
 
 
−
* [[베버(Weber) 모듈라 함수]]<br>
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
<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>
 
  
 
*  베버의 class invariant 라는 이름으로 잘 알려져 있으며, 베버는 Schläfli 함수로 불렀음<br>
 
*  베버의 class invariant 라는 이름으로 잘 알려져 있으며, 베버는 Schläfli 함수로 불렀음<br>
 
*  class field theory에서 중요한 역할<br>
 
*  class field theory에서 중요한 역할<br>
  
−
 
+
  
−
 
+
  
−
<h5 style="margin: 0px; line-height: 2em;">정의</h5>
+
==정의==
  
 
<math>\mathfrak{f}(\tau)=\frac{e^{-\frac{\pi i}{24}}\eta(\frac{\tau+1}{2})}{\eta(\tau)}=q^{-1/48} \prod_{n=1}^{\infty} (1+q^{n-\frac{1}{2}})</math>
 
<math>\mathfrak{f}(\tau)=\frac{e^{-\frac{\pi i}{24}}\eta(\frac{\tau+1}{2})}{\eta(\tau)}=q^{-1/48} \prod_{n=1}^{\infty} (1+q^{n-\frac{1}{2}})</math>
  
−
<math>\mathfrak{f}_1(\tau)=\frac{\eta(\frac{\tau}{2})}{\eta(\tau)}=q^{-1/48} \prod_{n=1}^{\infty} (1-q^{n-\frac{1}{2}})</math>
+
<math>\mathfrak{f}_ 1(\tau)=\frac{\eta(\frac{\tau}{2})}{\eta(\tau)}=q^{-1/48} \prod_{n=1}^{\infty} (1-q^{n-\frac{1}{2}})</math>
  
−
<math>\mathfrak{f}_2(\tau)=\sqrt{2}\frac{\eta(2\tau)}{\eta(\tau)}=\sqrt{2}q^{1/24} \prod_{n=1}^{\infty} (1+q^{n})</math>
+
<math>\mathfrak{f}_ 2(\tau)=\sqrt{2}\frac{\eta(2\tau)}{\eta(\tau)}=\sqrt{2}q^{1/24} \prod_{n=1}^{\infty} (1+q^{n})</math>
  
−
 
+
  
 
<math>\gamma_2(\tau)=\frac{\mathfrak{f}(\tau)^{24}-16}{\mathfrak{f}(\tau)^8}=\sqrt[3]{j(\tau)}</math>
 
<math>\gamma_2(\tau)=\frac{\mathfrak{f}(\tau)^{24}-16}{\mathfrak{f}(\tau)^8}=\sqrt[3]{j(\tau)}</math>
  
−
<math>\gamma_3(\tau)= \frac{(\mathfrak{f}(z)^{24} + 8) (\mathfrak{f}_1(z)^8 - \mathfrak{f}_2(z)^8)}{\mathfrak{f}(z)^8}=\sqrt{j(\tau)-1728}</math>
+
<math>\gamma_3(\tau)= \frac{(\mathfrak{f}(z)^{24} + 8) (\mathfrak{f}_ 1(z)^8 - \mathfrak{f}_ 2(z)^8)}{\mathfrak{f}(z)^8}=\sqrt{j(\tau)-1728}</math>
  
−
여기서  <math>\eta(\tau) = q^{1/24} \prod_{n=1}^{\infty} (1-q^{n})</math> 는 [[데데킨트 에타함수]]
+
여기서 <math>\eta(\tau) = q^{1/24} \prod_{n=1}^{\infty} (1-q^{n})</math> 는 [[데데킨트 에타함수]]
  
−
 
+
  
−
 
+
  
−
 
+
  
−
<h5 style="margin: 0px; line-height: 2em;">항등식</h5>
+
==항등식==
  
−
* <math>\mathfrak{f}_1(2\tau)\mathfrak{f}_2(\tau)=\sqrt2</math><br>
+
* <math>\mathfrak{f}_ 1(2\tau)\mathfrak{f}_ 2(\tau)=\sqrt2</math><br>
−
* <math>\mathfrak{f}(\tau)\mathfrak{f}_1(\tau)\mathfrak{f}_2(\tau)=\sqrt2</math><br>
+
* <math>\mathfrak{f}(\tau)\mathfrak{f}_ 1(\tau)\mathfrak{f}_ 2(\tau)=\sqrt2</math><br>
−
* <math>\mathfrak{f}(\tau)^8=\mathfrak{f}_1(\tau)^8+\mathfrak{f}_2(\tau)^8</math><br>
+
* <math>\mathfrak{f}(\tau)^8=\mathfrak{f}_ 1(\tau)^8+\mathfrak{f}_ 2(\tau)^8</math><br>
  
−
 
+
  
−
 
+
  
−
<h5 style="margin: 0px; line-height: 2em;">모듈라 성질</h5>
+
==모듈라 성질==
  
−
* <math>\mathfrak{f}(\tau+1)=\zeta_{48}^{-1}\mathfrak{f}_1(\tau)</math><br>
+
* <math>\mathfrak{f}(\tau+1)=\zeta_{48}^{-1}\mathfrak{f}_ 1(\tau)</math><br>
−
* <math>\mathfrak{f}_1(\tau+1)=\zeta_{48}^{-1}\mathfrak{f}(\tau)</math><br>
+
* <math>\mathfrak{f}_ 1(\tau+1)=\zeta_{48}^{-1}\mathfrak{f}(\tau)</math><br>
−
* <math>\mathfrak{f}_2(\tau+1)=\zeta_{24}\mathfrak{f}_2(\tau)</math><br>
+
* <math>\mathfrak{f}_ 2(\tau+1)=\zeta_{24}\mathfrak{f}_ 2(\tau)</math><br>
 
* <math>\mathfrak{f}(-\frac{1}{\tau})=\mathfrak{f}(\tau)</math><br>
 
* <math>\mathfrak{f}(-\frac{1}{\tau})=\mathfrak{f}(\tau)</math><br>
−
* <math>\mathfrak{f}_1(-\frac{1}{\tau})=\mathfrak{f}_2(\tau)</math><br>
+
* <math>\mathfrak{f}_ 1(-\frac{1}{\tau})=\mathfrak{f}_ 2(\tau)</math><br>
−
* <math>\mathfrak{f}_2(-\frac{1}{\tau})=\mathfrak{f}_1(\tau)</math><br>
+
* <math>\mathfrak{f}_ 2(-\frac{1}{\tau})=\mathfrak{f}_ 1(\tau)</math><br>
  
−
 
+
  
−
 
+
  
−
<h5>j-invariant 와의 관계</h5>
+
==j-invariant 와의 관계==
  
 
* [[타원 모듈라 j-함수 (elliptic modular function, j-invariant)|타원 모듈라 j-함수 (j-invariant)]]
 
* [[타원 모듈라 j-함수 (elliptic modular function, j-invariant)|타원 모듈라 j-함수 (j-invariant)]]
−
* <math>\mathfrak{f}(\tau)^{24}</math>, <math>-\mathfrak{f}_1(\tau)^{24}</math>, <math>-\mathfrak{f}_2(\tau)^{24}</math>는 <math>(x-16)^3-j(\tau)x=0</math> 의 근이다
+
* <math>\mathfrak{f}(\tau)^{24}</math>, <math>-\mathfrak{f}_ 1(\tau)^{24}</math>, <math>-\mathfrak{f}_ 2(\tau)^{24}</math>는 <math>(x-16)^3-j(\tau)x=0</math> 의 근이다
  
−
 
+
  
−
 
+
  
−
<h5 style="margin: 0px; line-height: 2em;">special values</h5>
+
==special values==
  
 
* <math>\mathfrak{f}(i)^8=4</math><br>
 
* <math>\mathfrak{f}(i)^8=4</math><br>
−
* <math>\mathfrak{f}_1(i)^8=2</math><br>
+
* <math>\mathfrak{f}_ 1(i)^8=2</math><br>
−
* <math>\mathfrak{f}_2(i)^8=2</math><br>
+
* <math>\mathfrak{f}_ 2(i)^8=2</math><br>
−
* <math>\mathfrak{f}_1(2i)^8=8</math><br>
+
* <math>\mathfrak{f}_ 1(2i)^8=8</math><br>
  
−
 
+
  
−
 
+
  
−
<h5 style="margin: 0px; line-height: 2em;">데데킨트 에타함수와의 관계</h5>
+
==데데킨트 에타함수와의 관계==
  
 
<math>\mathfrak{f}(\tau)=\frac{e^{-\frac{\pi i}{24}}\eta(\frac{\tau+1}{2})}{\eta(\tau)}=q^{-1/48} \prod_{n=1}^{\infty} (1+q^{n-\frac{1}{2}})</math>
 
<math>\mathfrak{f}(\tau)=\frac{e^{-\frac{\pi i}{24}}\eta(\frac{\tau+1}{2})}{\eta(\tau)}=q^{-1/48} \prod_{n=1}^{\infty} (1+q^{n-\frac{1}{2}})</math>
  
−
<math>\mathfrak{f}_1(\tau)=\frac{\eta(\frac{\tau}{2})}{\eta(\tau)}=q^{-1/48} \prod_{n=1}^{\infty} (1-q^{n-\frac{1}{2}})</math>
+
<math>\mathfrak{f}_ 1(\tau)=\frac{\eta(\frac{\tau}{2})}{\eta(\tau)}=q^{-1/48} \prod_{n=1}^{\infty} (1-q^{n-\frac{1}{2}})</math>
  
−
<math>\mathfrak{f}_2(\tau)=\sqrt{2}\frac{\eta(2\tau)}{\eta(\tau)}=\sqrt{2}q^{1/24} \prod_{n=1}^{\infty} (1+q^{n})</math>
+
<math>\mathfrak{f}_ 2(\tau)=\sqrt{2}\frac{\eta(2\tau)}{\eta(\tau)}=\sqrt{2}q^{1/24} \prod_{n=1}^{\infty} (1+q^{n})</math>
  
−
여기서  <math>\eta(\tau) = q^{1/24} \prod_{n=1}^{\infty} (1-q^{n})</math> 는 [[데데킨트 에타함수]]
+
여기서 <math>\eta(\tau) = q^{1/24} \prod_{n=1}^{\infty} (1-q^{n})</math> 는 [[데데킨트 에타함수]]
  
−
 
+
  
−
 
+
  
−
 
+
  
−
<h5 style="margin: 0px; line-height: 2em;">q-초기하급수와의 관계</h5>
+
==q-초기하급수와의 관계==
  
−
* [[q-초기하급수(q-hypergeometric series) (통합됨)|q-초기하급수(q-hypergeometric series)]] 의 공식<br><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><br><math>z=q^{1/2}</math> 인 경우<br><math>\prod_{n=1}^{\infty} (1+q^{n-\frac{1}{2}})=\sum_{n\geq 0}\frac{q^{n(n-1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)} (q^{1/2})^n=\sum_{n\geq 0}\frac{q^{n^2/2}}{(1-q)(1-q^2)\cdots(1-q^n)} </math><br><math>\prod_{n=1}^{\infty} (1+q^{2n-1})=\sum_{n\geq 0}\frac{q^{n^2}}{(1-q^2)(1-q^4)\cdots(1-q^{2n})} </math><br><math>z=q</math> 인 경우<br><math>\prod_{n=1}^{\infty} (1+q^{n})=\sum_{n\geq 0}\frac{q^{n(n-1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)}q^n=\sum_{n\geq 0}\frac{q^{n(n+1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)}</math><br>
+
* [[q-초기하급수(q-hypergeometric series) (통합됨)|q-초기하급수(q-hypergeometric series)]] 의 공식<br><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><br><math>z=q^{1/2}</math> 인 경우<br><math>\prod_{n=1}^{\infty} (1+q^{n-\frac{1}{2}})=\sum_{n\geq 0}\frac{q^{n (n-1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)} (q^{1/2})^n=\sum_{n\geq 0}\frac{q^{n^2/2}}{(1-q)(1-q^2)\cdots(1-q^n)} </math><br><math>\prod_{n=1}^{\infty} (1+q^{2n-1})=\sum_{n\geq 0}\frac{q^{n^2}}{(1-q^2)(1-q^4)\cdots(1-q^{2n})} </math><br><math>z=q</math> 인 경우<br><math>\prod_{n=1}^{\infty} (1+q^{n})=\sum_{n\geq 0}\frac{q^{n (n-1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)}q^n=\sum_{n\geq 0}\frac{q^{n (n+1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)}</math><br>
−
*  위의 결과로부터 다음을 얻을 수 있다<br><math>\mathfrak{f}(2\tau)=q^{-1/24}\prod_{n=1}^{\infty} (1+q^{2n-1})=q^{-1/24}\sum_{n\geq 0}\frac{q^{n^2}}{(1-q^2)(1-q^4)\cdots(1-q^{2n})}</math><br><math>\mathfrak{f}_2(\tau)=\sqrt{2}\frac{\eta(2\tau)}{\eta(\tau)}=\sqrt{2}q^{1/24} \prod_{n=1}^{\infty} (1+q^{n})=\sqrt{2}q^{1/24}\sum_{n\geq 0}\frac{q^{n(n+1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)}</math><br>
+
*  위의 결과로부터 다음을 얻을 수 있다<br><math>\mathfrak{f}(2\tau)=q^{-1/24}\prod_{n=1}^{\infty} (1+q^{2n-1})=q^{-1/24}\sum_{n\geq 0}\frac{q^{n^2}}{(1-q^2)(1-q^4)\cdots(1-q^{2n})}</math><br><math>\mathfrak{f}_ 2(\tau)=\sqrt{2}\frac{\eta(2\tau)}{\eta(\tau)}=\sqrt{2}q^{1/24} \prod_{n=1}^{\infty} (1+q^{n})=\sqrt{2}q^{1/24}\sum_{n\geq 0}\frac{q^{n (n+1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)}</math><br>
  
−
 
+
  
−
 
+
  
−
<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%;">역사</h5>
+
==역사==
  
 
* [[수학사연표 (역사)|수학사연표]]
 
* [[수학사연표 (역사)|수학사연표]]
  
−
 
+
  
−
 
+
  
−
<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%;">메모</h5>
+
==메모==
  
−
 
+
  
−
 
+
  
−
<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%;">관련된 항목들</h5>
+
==관련된 항목들==
  
 
* [[라마누잔의 class invariants]]<br>
 
* [[라마누잔의 class invariants]]<br>
129번째 줄: 121번째 줄:
 
* [[타원 모듈라 j-함수 (elliptic modular function, j-invariant)|타원 모듈라 j-함수 (j-invariant)]]<br>
 
* [[타원 모듈라 j-함수 (elliptic modular function, j-invariant)|타원 모듈라 j-함수 (j-invariant)]]<br>
  
−
 
+
  
−
 
+
  
−
<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>
+
==수학용어번역==
  
 
* http://www.google.com/dictionary?langpair=en|ko&q=
 
* http://www.google.com/dictionary?langpair=en|ko&q=
 
* [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= 대한수학회 수학 학술 용어집]<br>
 
** 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://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]
+
* [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]
  
−
 
+
  
−
 
+
  
−
<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%;">사전 형태의 자료</h5>
+
==사전 형태의 자료==
  
 
* http://ko.wikipedia.org/wiki/
 
* http://ko.wikipedia.org/wiki/
153번째 줄: 145번째 줄:
 
** http://www.research.att.com/~njas/sequences/?q=
 
** http://www.research.att.com/~njas/sequences/?q=
  
−
 
+
  
−
 
+
  
−
<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%;">관련논문</h5>
+
==관련논문==
  
 
* [http://jtnb.cedram.org/item?id=JTNB_2002__14_1_325_0 Weber's class invariants revisited]<br>
 
* [http://jtnb.cedram.org/item?id=JTNB_2002__14_1_325_0 Weber's class invariants revisited]<br>
−
** [http://jtnb.cedram.org/item?id=JTNB_2002__14_1_325_0 ]Reinhard Schertz, Journal de théorie des nombres de Bordeaux, 14 no. 1 (2002), p. 325-343 
+
** Reinhard Schertz, Journal de théorie des nombres de Bordeaux, 14 no. 1 (2002), p. 325-343
 
* [http://www.ams.org/mcom/1997-66-220/S0025-5718-97-00854-5/ On The Singular Values Of Weber Modular Functions]<br>
 
* [http://www.ams.org/mcom/1997-66-220/S0025-5718-97-00854-5/ On The Singular Values Of Weber Modular Functions]<br>
−
** Noriko Yui ,  Don Zagier, Math. Comp. 66 (1997), 1645-1662
+
** Noriko Yui , Don Zagier, Math. Comp. 66 (1997), 1645-1662
 
* [http://dx.doi.org/10.1112/S0025579300008251 Weber's Class Invariants]<br>
 
* [http://dx.doi.org/10.1112/S0025579300008251 Weber's Class Invariants]<br>
−
** B. J. Birch,  Mathematika 16 (1969)
+
** B. J. Birch, Mathematika 16 (1969)
 
* http://www.jstor.org/action/doBasicSearch?Query=
 
* http://www.jstor.org/action/doBasicSearch?Query=
  
−
 
+
  
−
 
+
  
−
<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%;">관련도서</h5>
+
==관련도서==
  
 
*  베버의 책<br>
 
*  베버의 책<br>
−
** [http://digital.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=06740001 Elliptische functionen und algebraische zahlen] (1891). [http://www.amazon.com/dp/1429701919?tag=corneunivelib-20 available in print]
+
** [http://digital.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=06740001 Elliptische functionen und algebraische zahlen] (1891). [http://www.amazon.com/dp/1429701919?tag=corneunivelib-20 available in print]
−
** [http://digital.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=webe031 Lehrbuch der Algebra (Volume 1)] (1898). [http://astech.library.cornell.edu/ast/math/additional/Digital-Books.cfm available in print]
+
** [http://digital.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=webe031 Lehrbuch der Algebra (Volume 1)] (1898). [http://astech.library.cornell.edu/ast/math/additional/Digital-Books.cfm available in print]
−
** [http://digital.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=webe032 Lehrbuch der Algebra (Volume 2)] (1898). [http://astech.library.cornell.edu/ast/math/additional/Digital-Books.cfm available in print]
+
** [http://digital.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=webe032 Lehrbuch der Algebra (Volume 2)] (1898). [http://astech.library.cornell.edu/ast/math/additional/Digital-Books.cfm available in print]
−
** [http://digital.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=webe034 Lehrbuch der Algebra (Volume 3)] (1898). [http://astech.library.cornell.edu/ast/math/additional/Digital-Books.cfm available in print]
+
** [http://digital.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=webe034 Lehrbuch der Algebra (Volume 3)] (1898). [http://astech.library.cornell.edu/ast/math/additional/Digital-Books.cfm available in print]
−
** [http://digital.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=02910001 Theorie der Abelschen Functionen vom Geschlecht 3] (1876). [http://www.amazon.com/dp/1429704683?tag=corneunivelib-20 available in print]
+
** [http://digital.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=02910001 Theorie der Abelschen Functionen vom Geschlecht 3] (1876). [http://www.amazon.com/dp/1429704683?tag=corneunivelib-20 available in print]

2012년 9월 10일 (월) 15:43 판

개요

  • 베버의 class invariant 라는 이름으로 잘 알려져 있으며, 베버는 Schläfli 함수로 불렀음
  • class field theory에서 중요한 역할



정의

\(\mathfrak{f}(\tau)=\frac{e^{-\frac{\pi i}{24}}\eta(\frac{\tau+1}{2})}{\eta(\tau)}=q^{-1/48} \prod_{n=1}^{\infty} (1+q^{n-\frac{1}{2}})\)

\(\mathfrak{f}_ 1(\tau)=\frac{\eta(\frac{\tau}{2})}{\eta(\tau)}=q^{-1/48} \prod_{n=1}^{\infty} (1-q^{n-\frac{1}{2}})\)

\(\mathfrak{f}_ 2(\tau)=\sqrt{2}\frac{\eta(2\tau)}{\eta(\tau)}=\sqrt{2}q^{1/24} \prod_{n=1}^{\infty} (1+q^{n})\)


\(\gamma_2(\tau)=\frac{\mathfrak{f}(\tau)^{24}-16}{\mathfrak{f}(\tau)^8}=\sqrt[3]{j(\tau)}\)

\(\gamma_3(\tau)= \frac{(\mathfrak{f}(z)^{24} + 8) (\mathfrak{f}_ 1(z)^8 - \mathfrak{f}_ 2(z)^8)}{\mathfrak{f}(z)^8}=\sqrt{j(\tau)-1728}\)

여기서 \(\eta(\tau) = q^{1/24} \prod_{n=1}^{\infty} (1-q^{n})\) 는 데데킨트 에타함수




항등식

  • \(\mathfrak{f}_ 1(2\tau)\mathfrak{f}_ 2(\tau)=\sqrt2\)
  • \(\mathfrak{f}(\tau)\mathfrak{f}_ 1(\tau)\mathfrak{f}_ 2(\tau)=\sqrt2\)
  • \(\mathfrak{f}(\tau)^8=\mathfrak{f}_ 1(\tau)^8+\mathfrak{f}_ 2(\tau)^8\)



모듈라 성질

  • \(\mathfrak{f}(\tau+1)=\zeta_{48}^{-1}\mathfrak{f}_ 1(\tau)\)
  • \(\mathfrak{f}_ 1(\tau+1)=\zeta_{48}^{-1}\mathfrak{f}(\tau)\)
  • \(\mathfrak{f}_ 2(\tau+1)=\zeta_{24}\mathfrak{f}_ 2(\tau)\)
  • \(\mathfrak{f}(-\frac{1}{\tau})=\mathfrak{f}(\tau)\)
  • \(\mathfrak{f}_ 1(-\frac{1}{\tau})=\mathfrak{f}_ 2(\tau)\)
  • \(\mathfrak{f}_ 2(-\frac{1}{\tau})=\mathfrak{f}_ 1(\tau)\)



j-invariant 와의 관계



special values

  • \(\mathfrak{f}(i)^8=4\)
  • \(\mathfrak{f}_ 1(i)^8=2\)
  • \(\mathfrak{f}_ 2(i)^8=2\)
  • \(\mathfrak{f}_ 1(2i)^8=8\)



데데킨트 에타함수와의 관계

\(\mathfrak{f}(\tau)=\frac{e^{-\frac{\pi i}{24}}\eta(\frac{\tau+1}{2})}{\eta(\tau)}=q^{-1/48} \prod_{n=1}^{\infty} (1+q^{n-\frac{1}{2}})\)

\(\mathfrak{f}_ 1(\tau)=\frac{\eta(\frac{\tau}{2})}{\eta(\tau)}=q^{-1/48} \prod_{n=1}^{\infty} (1-q^{n-\frac{1}{2}})\)

\(\mathfrak{f}_ 2(\tau)=\sqrt{2}\frac{\eta(2\tau)}{\eta(\tau)}=\sqrt{2}q^{1/24} \prod_{n=1}^{\infty} (1+q^{n})\)

여기서 \(\eta(\tau) = q^{1/24} \prod_{n=1}^{\infty} (1-q^{n})\) 는 데데킨트 에타함수




q-초기하급수와의 관계

  • q-초기하급수(q-hypergeometric series) 의 공식
    \(\prod_{n=0}^{\infty}(1+zq^n)=\sum_{n\geq 0}\frac{q^{n (n-1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)} z^n\)
    \(z=q^{1/2}\) 인 경우
    \(\prod_{n=1}^{\infty} (1+q^{n-\frac{1}{2}})=\sum_{n\geq 0}\frac{q^{n (n-1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)} (q^{1/2})^n=\sum_{n\geq 0}\frac{q^{n^2/2}}{(1-q)(1-q^2)\cdots(1-q^n)} \)
    \(\prod_{n=1}^{\infty} (1+q^{2n-1})=\sum_{n\geq 0}\frac{q^{n^2}}{(1-q^2)(1-q^4)\cdots(1-q^{2n})} \)
    \(z=q\) 인 경우
    \(\prod_{n=1}^{\infty} (1+q^{n})=\sum_{n\geq 0}\frac{q^{n (n-1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)}q^n=\sum_{n\geq 0}\frac{q^{n (n+1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)}\)
  • 위의 결과로부터 다음을 얻을 수 있다
    \(\mathfrak{f}(2\tau)=q^{-1/24}\prod_{n=1}^{\infty} (1+q^{2n-1})=q^{-1/24}\sum_{n\geq 0}\frac{q^{n^2}}{(1-q^2)(1-q^4)\cdots(1-q^{2n})}\)
    \(\mathfrak{f}_ 2(\tau)=\sqrt{2}\frac{\eta(2\tau)}{\eta(\tau)}=\sqrt{2}q^{1/24} \prod_{n=1}^{\infty} (1+q^{n})=\sqrt{2}q^{1/24}\sum_{n\geq 0}\frac{q^{n (n+1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)}\)



역사



메모

관련된 항목들



수학용어번역



사전 형태의 자료



관련논문



관련도서

원본 주소 "https://wiki.mathnt.net/index.php?title=베버(Weber)_모듈라_함수&oldid=10724"