"로그 사인 적분 (log sine integrals)"의 두 판 사이의 차이
| 91번째 줄: | 91번째 줄: | ||
* http://www.wolframalpha.com/input/?i=integrate+(log+(2sin(x/2)))^2+dx+from+0+to+pi/3<br> | * http://www.wolframalpha.com/input/?i=integrate+(log+(2sin(x/2)))^2+dx+from+0+to+pi/3<br> | ||
| − | * | + | * http://arxiv.org/abs/hep-ph/0411100v2<br> |
| 140번째 줄: | 140번째 줄: | ||
* Some wonderful formulas ... an introduction to polylogarithms<br> | * Some wonderful formulas ... an introduction to polylogarithms<br> | ||
** A.J. Van der Poorten, Queen's papers in Pure and Applied Mathematics, 54 (1979), 269-286 | ** A.J. Van der Poorten, Queen's papers in Pure and Applied Mathematics, 54 (1979), 269-286 | ||
| + | * [http://dx.doi.org/10.1007/BF01935325 On the value of a logarithmic-trigonometric integral]<br> | ||
| + | ** K. S. Kölbig, 1970 | ||
* [http://www.jstor.org/stable/3609410 On the Evaluation of log-sine Integrals]<br> | * [http://www.jstor.org/stable/3609410 On the Evaluation of log-sine Integrals]<br> | ||
** L. Lewin The Mathematical Gazette, Vol. 42, No. 340 (May, 1958), pp. 125-128<br> | ** L. Lewin The Mathematical Gazette, Vol. 42, No. 340 (May, 1958), pp. 125-128<br> | ||
| − | * | + | * [http://dx.doi.org/10.1112/jlms/s1-22.3.172 Note on the Integral] <br> |
| − | ** J. London Math. Soc. 1947 s1-22: 172-173<br> | + | ** F. Bowman, J. London Math. Soc. 1947 s1-22: 172-173<br> |
* http://www.jstor.org/stable/3609410 | * http://www.jstor.org/stable/3609410 | ||
* http://www.jstor.org/action/doBasicSearch?Query= | * http://www.jstor.org/action/doBasicSearch?Query= | ||
* http://www.ams.org/mathscinet | * http://www.ams.org/mathscinet | ||
| − | * http://dx.doi.org/ | + | * http://dx.doi.org/10.1007/BF01935325 |
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2010년 6월 15일 (화) 21:31 판
이 항목의 스프링노트 원문주소
개요
- 정의
\(\operatorname{Ls}_{a+b,a}(\theta)=-\int_{0}^{\theta}x^a\log^{b-1}}|2\sin \frac{x}{2}|\,dx\)
- 클라우센 함수의 일반화로 볼 수 있다
\(\operatorname{Cl}_2(\theta)=-\int_0^{\theta} \ln |2\sin \frac{t}{2}| \,dt=\sum_{n=1}^{\infty}\frac{\sin (n\theta)}{n^2}\)
\(\int_{0}^{1-e^{i\theta}}\log^{n-1}z\frac{dz}{1-z}=-i\int_{0}^{\theta}(\frac{i}{2}(x-\pi)+\log|2\sin \frac{x}{2}|)^{n-1}\,dx \)\(=-\int_{0}^{\theta}x^a\log^{b-1}}|2\sin \frac{x}{2}|\,dx\)
special values의 생성함수
- 정의
\(\operatorname{Ls}_{n}(\pi)=-\int_{0}^{\pi}\log^{n-1}}(2\sin \frac{x}{2})\,dx\) - 생성함수
\(I(x)=\int_{0}^{\pi}e^{x\log(2\sin \frac{1}{2}\theta)}d\theta =\sum_{n=0}^{\infty}\int_{0}^{\pi}\frac{x^n}{n!}\log^n(2\sin\frac{1}{2}\theta)d\theta=-\sum_{n=0}^{\infty}\frac{x^n}{n!}\operatorname{Ls}_{n+1}(\pi)\)
\(I(x)=\frac{\pi\Gamma(1+x)}{(\Gamma(1+\frac{1}{2}x))^2}\)
점화식
\(\operatorname{Ls}_{m+2}(\pi)=(-1)^{m}m[\pi(1-2^{-m})\zeta(m+1)-(1-2^{2-m})\zeta(m-1)\operatorname{Ls}_{3}(\pi)/2!+(1-2^{3-m})\zeta(m-2)\operatorname{Ls}_{4}(\pi)/3\cdots+!+(-1)^{m}(1-1/2})\zeta(2)\operatorname{Ls}_{m}(\pi)/(m-1)!]\)
special values
\(\int_{0}^{\pi/2}\log(\sin x)\,dx=-\frac{\pi\log 2}{2}\)
\(\int_{0}^{\pi/2}\log^2(\sin x)\,dx=\frac{\pi}{2}(\log 2)^2+\frac{\pi^3}{24}\)
\(\operatorname{Ls}_2(\pi)=-\int_{0}^{\pi}\log(2\sin \frac{x}{2})\,dx=0\)
\(\operatorname{Ls}_3(\pi)=-\int_{0}^{\pi}\log^2(2\sin \frac{x}{2})\,dx=-\frac{\pi^3}{12}\)
\(\operatorname{Ls}_4(\pi)=-\int_{0}^{\pi}\log^3(2\sin \frac{x}{2})\,dx=\frac{3\pi}{2}\zeta(3)\)
\(\operatorname{Ls}_5(\pi)=-\int_{0}^{\pi}\log^4(2\sin \frac{x}{2})\,dx=-\frac{19\pi^5}{240}\)
\(\operatorname{Ls}_6(\pi)=-\int_{0}^{\pi}\log^5(2\sin \frac{x}{2})\,dx=\frac{45\pi}{2}\zeta(5)+\frac{5\pi^3}{4}\zeta(3)\)\(\operatorname{Ls}_7(\pi)=-\int_{0}^{\pi}\log^6(2\sin \frac{x}{2})\,dx=-\frac{45\pi}{2}\zeta^2(3)-\frac{275\pi^7}{1344}\)
\(\int_{0}^{\pi/3}\log^2(2\sin \frac{x}{2})\,dx=\frac{7\pi^3}{108}\)
\(\int_{0}^{\pi/3}x\log^2(2\sin \frac{x}{2})\,dx=\frac{17\pi^4}{6480}\)
재미있는 사실
- Math Overflow http://mathoverflow.net/search?q=
- 네이버 지식인 http://kin.search.naver.com/search.naver?where=kin_qna&query=
역사
- http://www.google.com/search?hl=en&tbs=tl:1&q=log+sine+integral
- http://www.google.com/search?hl=en&tbs=tl:1&q=
- 수학사연표
메모
- http://www.wolframalpha.com/input/?i=integrate+(log+(2sin(x/2)))^2+dx+from+0+to+pi/3
- http://arxiv.org/abs/hep-ph/0411100v2
관련된 항목들
수학용어번역
- 단어사전 http://www.google.com/dictionary?langpair=en%7Cko&q=
- 발음사전 http://www.forvo.com/search/
- 대한수학회 수학 학술 용어집
- 남·북한수학용어비교
- 대한수학회 수학용어한글화 게시판
사전 형태의 자료
- http://ko.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/
- http://www.wolframalpha.com/input/?i=
- NIST Digital Library of Mathematical Functions
- The On-Line Encyclopedia of Integer Sequences
관련논문
- Some wonderful formulas ... an introduction to polylogarithms
- A.J. Van der Poorten, Queen's papers in Pure and Applied Mathematics, 54 (1979), 269-286
- On the value of a logarithmic-trigonometric integral
- K. S. Kölbig, 1970
- On the Evaluation of log-sine Integrals
- L. Lewin The Mathematical Gazette, Vol. 42, No. 340 (May, 1958), pp. 125-128
- L. Lewin The Mathematical Gazette, Vol. 42, No. 340 (May, 1958), pp. 125-128
- Note on the Integral
- F. Bowman, J. London Math. Soc. 1947 s1-22: 172-173
- F. Bowman, J. London Math. Soc. 1947 s1-22: 172-173
- http://www.jstor.org/stable/3609410
- http://www.jstor.org/action/doBasicSearch?Query=
- http://www.ams.org/mathscinet
- http://dx.doi.org/10.1007/BF01935325
관련도서
- 도서내검색
- 도서검색
관련기사
- 네이버 뉴스 검색 (키워드 수정)