"클라우센 함수(Clausen function)"의 두 판 사이의 차이

수학노트
둘러보기로 가기 검색하러 가기
잔글 (찾아 바꾸기 – “==관련기사== * 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=)
 
(같은 사용자의 중간 판 8개는 보이지 않습니다)
1번째 줄: 1번째 줄:
−
==이 항목의 스프링노트 원문주소==
 
−
 
−
* [[클라우센 함수(Clausen function)]]<br>
 
−
 
−
 
 
−
 
−
 
 
−
 
 
==개요==
 
==개요==
  
−
*  정의<br><math>\operatorname{Cl}_2(\theta)=-\int_0^{\theta} \ln |2\sin \frac{t}{2}| \,dt=\sum_{n=1}^{\infty}\frac{\sin (n\theta)}{n^2}</math><br>
+
*  정의:<math>\operatorname{Cl}_2(\theta)=-\int_0^{\theta} \ln |2\sin \frac{t}{2}| \,dt=\sum_{n=1}^{\infty}\frac{\sin (n\theta)}{n^2}</math>
  
−
* [[로그 사인 적분 (log sine integrals)]] 으로 일반화된다<br>
+
* [[로그 사인 적분 (log sine integrals)]] 으로 일반화된다
−
* [[로바체프스키 함수]] 와의 관계<br><math>Cl_2(2\theta)=2\Lambda(\theta)</math><br>
+
* [[로바체프스키 함수]] 와의 관계:<math>\operatorname{Cl}_2(2\theta)=2\Lambda(\theta)</math>
  
−
 
 
  
−
 
+
  
−
 
+
==다이로그 함수와의 관계==
  
−
==dilogarithm 함수와의 관계==
+
* [[다이로그 함수(dilogarithm)]]는 복소수 <math>|z|<1</math>에 대하여 다음과 같이 정의됨
 +
:<math>\operatorname{Li}_2(z)= \sum_{n=1}^\infty {z^n \over n^2}</math>
 +
* <math>|z|\leq 1</math> 에서 고르게 수렴하는 급수이므로, <math>|z|\leq 1</math>에서 연속
 +
* <math>z=e^{i\theta}</math>, <math>0 \leq \theta \leq 2\pi</math> 일때:<math>\operatorname{Li}_2(e^{i\theta})= \sum_{n=1}^\infty \frac{e^{in\theta}}{n^2}=\sum_{n=1}^\infty \frac{\cos n\theta}{n^2}+i\sum_{n=1}^\infty \frac{\sin n\theta}{n^2}</math>:<math>\mathfrak{I}(\operatorname{Li}_2(e^{i\theta}))=\sum_{n=1}^\infty \frac{\sin n\theta}{n^2}=\operatorname{Cl}_2(\theta)</math>
 +
* [[블로흐-비그너 다이로그(Bloch-Wigner dilogarithm)]]
  
−
* [[#]]
+
   
−
* 는 복소수 <math>|z|<1</math>에 대하여 다음과 같이 정의됨
 
−
* <math>\operatorname{Li}_2(z)= \sum_{n=1}^\infty {z^n \over n^2}</math><br><math>|z|\leq 1</math> 에서 고르게 수렴하는 급수이므로, <math>|z|\leq 1</math>에서 연속<br>
 
−
*  <br><math>z=e^{i\theta}</math>, <math>0 \leq \theta \leq 2\pi</math> 일때<br><math>\operatorname{Li}_2(e^{i\theta})= \sum_{n=1}^\infty \frac{e^{in\theta}}{n^2}=\sum_{n=1}^\infty \frac{\cos n\theta}{n^2}+i\sum_{n=1}^\infty \frac{\sin n\theta}{n^2}</math><br><math>\mathfrak{I}(\operatorname{Li}_2(e^{i\theta}))=\sum_{n=1}^\infty \frac{\sin n\theta}{n^2}=Cl_2(\theta)</math>[[블로흐-비그너 다이로그(Bloch-Wigner dilogarithm)|블로흐-비그너 다이로그]]<br>
 
−
* [[블로흐-비그너 다이로그(Bloch-Wigner dilogarithm)]]<br>
 
  
−
 
+
−
 
 
−
 
 
  
 
==덧셈공식==
 
==덧셈공식==
36번째 줄: 25번째 줄:
 
<math>\Lambda(n\theta)=n\sum_{k=0}^{n-1}\Lambda(\theta+\frac{2k\pi}{n})</math>
 
<math>\Lambda(n\theta)=n\sum_{k=0}^{n-1}\Lambda(\theta+\frac{2k\pi}{n})</math>
  
−
 
+
  
−
 
+
  
−
==트리감마 함수와 special values==
+
==트리감마 함수와 special values==
  
−
* <math>\theta=p\pi/q</math>일 때, (<math>p,q\in\mathbb{N}</math>, <math>p=1,2,\cdots,2q-1</math>)<br><math>\operatorname{Cl}_2(\frac{p\pi}{q})=\frac{1}{4q^2}\sum_{r=1}^{2q-1}\psi^{(1)}(\frac{r}{2q})\sin\frac{rp\pi}{q}</math><br> 여기서 <math>\psi^{(1)}</math>는 [[트리감마 함수(trigamma function)]]<br>
+
* <math>\theta=p\pi/q</math>일 때, (<math>p,q\in\mathbb{N}</math>, <math>p=1,2,\cdots,2q-1</math>):<math>\operatorname{Cl}_2(\frac{p\pi}{q})=\frac{1}{4q^2}\sum_{r=1}^{2q-1}\psi^{(1)}(\frac{r}{2q})\sin\frac{rp\pi}{q}</math> 여기서 <math>\psi^{(1)}</math>는 [[트리감마 함수(trigamma function)]]
−
* <math>\operatorname{Cl}_2(\frac{\pi}{2})=G</math>, <math>G</math>는 [[카탈란 상수]]<br>
+
* <math>\operatorname{Cl}_2(\frac{\pi}{2})=G</math>, <math>G</math>는 [[카탈란 상수]]
−
* <math>\operatorname{Cl}_2(\frac{\pi}{3})=\frac{\sqrt{3}}{12}(\psi^{(1)}(\frac{1}{3})-\psi^{(1)}(\frac{2}{3}))</math><br>
+
* <math>\operatorname{Cl}_2(\frac{\pi}{3})=\frac{\sqrt{3}}{12}(\psi^{(1)}(\frac{1}{3})-\psi^{(1)}(\frac{2}{3}))</math>
−
* http://mathworld.wolfram.com/GiesekingsConstant.html<br>
+
* http://mathworld.wolfram.com/GiesekingsConstant.html
  
−
 
+
  
−
 
+
−
 
 
−
==재미있는 사실==
 
−
 
 
−
 
 
−
 
 
−
* Math Overflow http://mathoverflow.net/search?q=
 
−
* 네이버 지식인 http://kin.search.naver.com/search.naver?where=kin_qna&query=
 
−
 
 
−
 
 
−
 
 
−
 
 
  
 
==역사==
 
==역사==
  
−
 
+
  
 
* http://www.google.com/search?hl=en&tbs=tl:1&q=
 
* http://www.google.com/search?hl=en&tbs=tl:1&q=
−
* [[수학사연표 (역사)|수학사연표]]
+
* [[수학사 연표]]
−
*  
+
 
  
−
 
+
  
−
 
+
  
 
==메모==
 
==메모==
78번째 줄: 56번째 줄:
 
<math>\int_{0}^{\pi/3}\operatorname{Cl}_2(x)\,dx=\frac{2}{3}\zeta(3)</math>
 
<math>\int_{0}^{\pi/3}\operatorname{Cl}_2(x)\,dx=\frac{2}{3}\zeta(3)</math>
  
−
 
+
  
 
==관련된 항목들==
 
==관련된 항목들==
  
−
* [[트리감마 함수(trigamma function)]]<br>
+
* [[트리감마 함수(trigamma function)]]
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
==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]<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]
 
−
 
 
−
 
 
  
−
 
+
  
−
==사전 형태의 자료==
+
==사전 형태의 자료==
  
 
* http://ko.wikipedia.org/wiki/
 
* http://ko.wikipedia.org/wiki/
 
* http://en.wikipedia.org/wiki/Clausen_function
 
* http://en.wikipedia.org/wiki/Clausen_function
 
* http://mathworld.wolfram.com/ClausensIntegral.html
 
* http://mathworld.wolfram.com/ClausensIntegral.html
−
* http://www.wolframalpha.com/input/?i=
 
−
* [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]<br>
 
−
** http://www.research.att.com/~njas/sequences/?q=
 
  
−
 
 
  
−
 
+
  
 
==관련논문==
 
==관련논문==
  
−
* [http://link.aip.org/link/?JMAPAQ/50/023515/1 A dilogarithmic integral arising in quantum field theory]<br>
+
* [http://link.aip.org/link/?JMAPAQ/50/023515/1 A dilogarithmic integral arising in quantum field theory]
 
** Djurdje Cvijović, J. Math. Phys. 50, 023515 (2009)
 
** Djurdje Cvijović, J. Math. Phys. 50, 023515 (2009)
−
* [http://link.aip.org/link/?JMAPAQ/49/043510/1 On a three-dimensional symmetric Ising tetrahedron and contributions to the theory of the dilogarithm and Clausen functions]<br>
+
* [http://link.aip.org/link/?JMAPAQ/49/043510/1 On a three-dimensional symmetric Ising tetrahedron and contributions to the theory of the dilogarithm and Clausen functions]
 
** Mark W. Coffey, J. Math. Phys. 49, 043510 (2008); doi:10.1063/1.2902996
 
** Mark W. Coffey, J. Math. Phys. 49, 043510 (2008); doi:10.1063/1.2902996
−
* [http://link.aip.org/link/?JMAPAQ/49/093508/1 Evaluation of a ln tan integral arising in quantum field theory]<br>
+
* [http://link.aip.org/link/?JMAPAQ/49/093508/1 Evaluation of a ln tan integral arising in quantum field theory]
 
** Mark W. Coffey, J. Math. Phys. 49, 093508 (2008); doi:10.1063/1.2981311
 
** Mark W. Coffey, J. Math. Phys. 49, 093508 (2008); doi:10.1063/1.2981311
−
* [http://www.sciencedirect.com/science?_ob=MathURL&_method=retrieve&_udi=B6TYH-4FM01DY-1&_mathId=mml206&_user=4420&_cdi=5619&_pii=S0377042705000154&_rdoc=1&_issn=03770427&_acct=C000059607&_version=1&_userid=4420&md5=6b9447739891f5d15aa022b55b859ef5 ][http://dx.doi.org/10.1016/0377-0427%2884%2990008-6 Formulae concerning the computation of the Clausen integral Cl2(θ)]<br>
+
* [http://dx.doi.org/10.1016/0377-0427%2884%2990008-6 Formulae concerning the computation of the Clausen integral Cl2(θ)]
−
** C.C. Grosjean, <em style="line-height: 2em;">J. Comput. Appl. Math.</em> '''11''' (1984), pp. 331–342
+
** C.C. Grosjean, <em style="line-height: 2em;">J. Comput. Appl. Math.</em> '''11''' (1984), pp. 331–342
−
* [http://dx.doi.org/10.1016/0377-0427%2884%2990007-4 On the Clausen integral Cl2(Θ) and a related integral]<br>
+
* [http://dx.doi.org/10.1016/0377-0427%2884%2990007-4 On the Clausen integral Cl2(Θ) and a related integral]
 
** P. J. de Doelder, J. Comput. Appl. Math. 11, 325 (1984)
 
** P. J. de Doelder, J. Comput. Appl. Math. 11, 325 (1984)
−
* [http://www.jstor.org/stable/2004590 Efficient Calculation of Clausen's Integral]<br>
+
* [http://www.jstor.org/stable/2004590 Efficient Calculation of Clausen's Integral]
 
** Van E. Wood, Mathematics of Computation, Vol. 22, No. 104 (Oct., 1968), pp. 883-884
 
** Van E. Wood, Mathematics of Computation, Vol. 22, No. 104 (Oct., 1968), pp. 883-884
  
−
* http://www.jstor.org/action/doBasicSearch?Query=
+
[[분류:다이로그]]
−
* http://www.ams.org/mathscinet
 
−
* http://dx.doi.org/
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
  
−
==블로그==
+
==메타데이터==
 +
===위키데이터===
 +
* ID :  [https://www.wikidata.org/wiki/Q823290 Q823290]
 +
===Spacy 패턴 목록===
 +
* [{'LOWER': 'clausen'}, {'LEMMA': 'function'}]

2021년 2월 17일 (수) 05:02 기준 최신판

개요

  • 정의\[\operatorname{Cl}_2(\theta)=-\int_0^{\theta} \ln |2\sin \frac{t}{2}| \,dt=\sum_{n=1}^{\infty}\frac{\sin (n\theta)}{n^2}\]



다이로그 함수와의 관계

\[\operatorname{Li}_2(z)= \sum_{n=1}^\infty {z^n \over n^2}\]

  • \(|z|\leq 1\) 에서 고르게 수렴하는 급수이므로, \(|z|\leq 1\)에서 연속
  • \(z=e^{i\theta}\), \(0 \leq \theta \leq 2\pi\) 일때\[\operatorname{Li}_2(e^{i\theta})= \sum_{n=1}^\infty \frac{e^{in\theta}}{n^2}=\sum_{n=1}^\infty \frac{\cos n\theta}{n^2}+i\sum_{n=1}^\infty \frac{\sin n\theta}{n^2}\]\[\mathfrak{I}(\operatorname{Li}_2(e^{i\theta}))=\sum_{n=1}^\infty \frac{\sin n\theta}{n^2}=\operatorname{Cl}_2(\theta)\]
  • 블로흐-비그너 다이로그(Bloch-Wigner dilogarithm)



덧셈공식

\(\Lambda(n\theta)=n\sum_{k=0}^{n-1}\Lambda(\theta+\frac{2k\pi}{n})\)



트리감마 함수와 special values



역사




메모

\(\int_{0}^{\pi/3}\operatorname{Cl}_2(x)\,dx=\frac{2}{3}\zeta(3)\)


관련된 항목들


사전 형태의 자료



관련논문

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'clausen'}, {'LEMMA': 'function'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=클라우센_함수(Clausen_function)&oldid=52499"