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

수학노트
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==개요==
 
==개요==
  
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*  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><br>
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*  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>
  
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* [[로그 사인 적분 (log sine integrals)]] 으로 일반화된다<br>
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* [[로그 사인 적분 (log sine integrals)]] 으로 일반화된다
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* [[로바체프스키 함수]] 와의 관계:<math>Cl_2(2\theta)=2\Lambda(\theta)</math><br>
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* [[로바체프스키 함수]] 와의 관계:<math>\operatorname{Cl}_2(2\theta)=2\Lambda(\theta)</math>
  
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==다이로그 함수와의 관계==
  
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==dilogarithm 함수와의 관계==
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* [[다이로그 함수(dilogarithm)]]는 복소수 <math>|z|<1</math>에 대하여 다음과 같이 정의됨
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* [[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>\operatorname{Li}_2(z)= \sum_{n=1}^\infty {z^n \over n^2}</math>
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* <math>|z|\leq 1</math> 에서 고르게 수렴하는 급수이므로, <math>|z|\leq 1</math>에서 연속
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* <math>|z|\leq 1</math> 에서 고르게 수렴하는 급수이므로, <math>|z|\leq 1</math>에서 연속
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* <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}=Cl_2(\theta)</math>
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* <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>
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* [[블로흐-비그너 다이로그(Bloch-Wigner dilogarithm)]]<br>
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* [[블로흐-비그너 다이로그(Bloch-Wigner dilogarithm)]]
  
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==덧셈공식==
 
==덧셈공식==
28번째 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>
  
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==트리감마 함수와 special values==
 
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* <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><br> 여기서 <math>\psi^{(1)}</math>는 [[트리감마 함수(trigamma function)]]<br>
 
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* <math>\operatorname{Cl}_2(\frac{\pi}{2})=G</math>, <math>G</math>는 [[카탈란 상수]]<br>
 
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* <math>\operatorname{Cl}_2(\frac{\pi}{3})=\frac{\sqrt{3}}{12}(\psi^{(1)}(\frac{1}{3})-\psi^{(1)}(\frac{2}{3}))</math><br>
 
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* http://mathworld.wolfram.com/GiesekingsConstant.html<br>
 
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==재미있는 사실==
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==트리감마 함수와 special values==
  
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* Math Overflow http://mathoverflow.net/search?q=
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* <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)]]
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* 네이버 지식인 http://kin.search.naver.com/search.naver?where=kin_qna&query=
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* <math>\operatorname{Cl}_2(\frac{\pi}{2})=G</math>, <math>G</math>는 [[카탈란 상수]]
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* <math>\operatorname{Cl}_2(\frac{\pi}{3})=\frac{\sqrt{3}}{12}(\psi^{(1)}(\frac{1}{3})-\psi^{(1)}(\frac{2}{3}))</math>
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* http://mathworld.wolfram.com/GiesekingsConstant.html
  
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==역사==
 
==역사==
  
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
 
* http://www.google.com/search?hl=en&tbs=tl:1&q=
 
* [[수학사 연표]]
 
* [[수학사 연표]]
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==메모==
 
==메모==
70번째 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>
  
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==관련된 항목들==
 
==관련된 항목들==
  
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* [[트리감마 함수(trigamma function)]]<br>
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* [[트리감마 함수(trigamma function)]]
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==수학용어번역==
 
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* 단어사전 http://www.google.com/dictionary?langpair=en|ko&q=
 
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* 발음사전 http://www.forvo.com/search/
 
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* [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집]<br>
 
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** http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
 
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* [http://www.nktech.net/science/term/term_l.jsp?l_mode=cate&s_code_cd=MA 남·북한수학용어비교]
 
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* [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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==사전 형태의 자료==
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==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
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* http://www.wolframalpha.com/input/?i=
 
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* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
 
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* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]<br>
 
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** http://www.research.att.com/~njas/sequences/?q=
 
  
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==관련논문==
 
==관련논문==
  
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* [http://link.aip.org/link/?JMAPAQ/50/023515/1 A dilogarithmic integral arising in quantum field theory]<br>
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* [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)
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* [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>
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* [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
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* [http://link.aip.org/link/?JMAPAQ/49/093508/1 Evaluation of a ln tan integral arising in quantum field theory]<br>
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* [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
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* [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>
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* [http://dx.doi.org/10.1016/0377-0427%2884%2990008-6 Formulae concerning the computation of the Clausen integral Cl2(θ)]
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** C.C. Grosjean, <em style="line-height: 2em;">J. Comput. Appl. Math.</em> '''11''' (1984), pp. 331–342
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** C.C. Grosjean, <em style="line-height: 2em;">J. Comput. Appl. Math.</em> '''11''' (1984), pp. 331–342
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* [http://dx.doi.org/10.1016/0377-0427%2884%2990007-4 On the Clausen integral Cl2(Θ) and a related integral]<br>
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* [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)
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* [http://www.jstor.org/stable/2004590 Efficient Calculation of Clausen's Integral]<br>
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* [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
  
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* http://www.jstor.org/action/doBasicSearch?Query=
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[[분류:다이로그]]
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* http://www.ams.org/mathscinet
 
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* http://dx.doi.org/
 
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==메타데이터==
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===위키데이터===
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==블로그==
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* ID :  [https://www.wikidata.org/wiki/Q823290 Q823290]
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[[분류:다이로그]]
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===Spacy 패턴 목록===
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* [{'LOWER': 'clausen'}, {'LEMMA': 'function'}]

2021년 2월 17일 (수) 06:02 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"