"클라우센 함수(Clausen function)"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) |
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| (사용자 2명의 중간 판 21개는 보이지 않습니다) | |||
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| − | + | ==개요== | |
| − | * | + | * 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 | |
| + | * [[로바체프스키 함수]] 와의 관계:<math>\operatorname{Cl}_2(2\theta)=2\Lambda(\theta)</math> | ||
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| − | + | ==다이로그 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)]] | ||
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| − | + | ==덧셈공식== | |
| − | < | + | <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== | |
| − | + | * <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]] | ||
| + | * <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 | ||
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| − | + | ==역사== | |
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| − | + | * http://www.google.com/search?hl=en&tbs=tl:1&q= | |
| − | + | * [[수학사 연표]] | |
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| − | + | ==메모== | |
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<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== | |
| − | + | * [[트리감마 함수(trigamma function)]] | |
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| − | + | ==사전 형태의 자료== | |
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* 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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| − | + | ==관련논문== | |
| − | * [http://link.aip.org/link/?JMAPAQ/50/023515/1 A dilogarithmic integral arising in quantum field theory] | + | * [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] | + | * [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] | + | * [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://dx.doi.org/10.1016/0377-0427%2884%2990008-6 Formulae concerning the computation of the Clausen integral Cl2(θ)] |
| − | ** C.C. Grosjean, | + | ** 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] | + | * [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] | + | * [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 | ||
| − | + | [[분류:다이로그]] | |
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| − | + | ==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\[\operatorname{Cl}_2(\theta)=-\int_0^{\theta} \ln |2\sin \frac{t}{2}| \,dt=\sum_{n=1}^{\infty}\frac{\sin (n\theta)}{n^2}\]
- 로그 사인 적분 (log sine integrals) 으로 일반화된다
- 로바체프스키 함수 와의 관계\[\operatorname{Cl}_2(2\theta)=2\Lambda(\theta)\]
다이로그 함수와의 관계
- 다이로그 함수(dilogarithm)는 복소수 \(|z|<1\)에 대하여 다음과 같이 정의됨
\[\operatorname{Li}_2(z)= \sum_{n=1}^\infty {z^n \over n^2}\]
- \(|z|\leq 1\) 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
- \(\theta=p\pi/q\)일 때, (\(p,q\in\mathbb{N}\), \(p=1,2,\cdots,2q-1\))\[\operatorname{Cl}_2(\frac{p\pi}{q})=\frac{1}{4q^2}\sum_{r=1}^{2q-1}\psi^{(1)}(\frac{r}{2q})\sin\frac{rp\pi}{q}\] 여기서 \(\psi^{(1)}\)는 트리감마 함수(trigamma function)
- \(\operatorname{Cl}_2(\frac{\pi}{2})=G\), \(G\)는 카탈란 상수
- \(\operatorname{Cl}_2(\frac{\pi}{3})=\frac{\sqrt{3}}{12}(\psi^{(1)}(\frac{1}{3})-\psi^{(1)}(\frac{2}{3}))\)
- http://mathworld.wolfram.com/GiesekingsConstant.html
역사
메모
\(\int_{0}^{\pi/3}\operatorname{Cl}_2(x)\,dx=\frac{2}{3}\zeta(3)\)
관련된 항목들
사전 형태의 자료
- http://ko.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/Clausen_function
- http://mathworld.wolfram.com/ClausensIntegral.html
관련논문
- A dilogarithmic integral arising in quantum field theory
- Djurdje Cvijović, J. Math. Phys. 50, 023515 (2009)
- 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
- 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
- Formulae concerning the computation of the Clausen integral Cl2(θ)
- C.C. Grosjean, J. Comput. Appl. Math. 11 (1984), pp. 331–342
- On the Clausen integral Cl2(Θ) and a related integral
- P. J. de Doelder, J. Comput. Appl. Math. 11, 325 (1984)
- Efficient Calculation of Clausen's Integral
- Van E. Wood, Mathematics of Computation, Vol. 22, No. 104 (Oct., 1968), pp. 883-884
메타데이터
위키데이터
- ID : Q823290
Spacy 패턴 목록
- [{'LOWER': 'clausen'}, {'LEMMA': 'function'}]