"조화수열과 조화급수"의 두 판 사이의 차이
Pythagoras0 (토론 | 기여) |
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− | + | ==개요== | |
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* 조화수열의 정의<br><math>H_{n}=\sum_{k=1}^{n}\frac{1}{k}</math><br> | * 조화수열의 정의<br><math>H_{n}=\sum_{k=1}^{n}\frac{1}{k}</math><br> | ||
− | * [[오일러상수, 감마]][[오일러상수, 감마|]]<math>\lim_{n\to\infty}H_{n}-\ln n=\gamma</math><br> | + | * [[오일러상수, 감마]][[오일러상수, 감마|오일러상수]]<math>\lim_{n\to\infty}H_{n}-\ln n=\gamma</math><br> |
<math>\gamma=0.577215664901532860606512090\cdots</math> | <math>\gamma=0.577215664901532860606512090\cdots</math> | ||
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− | <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px;">근사 공식 | + | <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px;">근사 공식== |
* [[오일러-맥클로린 공식]] 을 통해 다음을 얻는다<br><math>H_{n}=\sum_{k=1}^{n}\frac{1}{k}\sim \log n +\gamma+\frac{1}{2n}-\sum_{s=1}^{\infty}\frac{B_{2s}}{(2s)n^{2s}}</math><br> | * [[오일러-맥클로린 공식]] 을 통해 다음을 얻는다<br><math>H_{n}=\sum_{k=1}^{n}\frac{1}{k}\sim \log n +\gamma+\frac{1}{2n}-\sum_{s=1}^{\infty}\frac{B_{2s}}{(2s)n^{2s}}</math><br> | ||
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− | + | ==성질== | |
<math>H_{n-1}=H_n-\frac{1}{n}</math> | <math>H_{n-1}=H_n-\frac{1}{n}</math> | ||
− | <math>H_{n-1}^2=(H_n-\frac{1}{n})^2=H_n^2+\frac{1}{n^2}-\frac{2H_n}{n}</math> | + | <math>H_ {n-1}^2=(H_n-\frac{1}{n})^2=H_n^2+\frac{1}{n^2}-\frac{2H_n}{n}</math> |
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− | + | ==생성함수== | |
<math>\sum_{n=1}^\infty H_nz^n = \frac {-\ln(1-z)}{1-z}</math> | <math>\sum_{n=1}^\infty H_nz^n = \frac {-\ln(1-z)}{1-z}</math> | ||
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− | + | ==생성함수의 응용== | |
<math>\sum_{n=1}^\infty \frac{H_n}{n+1}z^{n+1} =\frac{1}{2}\log^2(1-z)</math> | <math>\sum_{n=1}^\infty \frac{H_n}{n+1}z^{n+1} =\frac{1}{2}\log^2(1-z)</math> | ||
− | <math>\sum_{n=1}^\infty \frac{H_n}{n}z^n =\operatorname{Li} | + | <math>\sum_{n=1}^\infty \frac{H_n}{n}z^n =\operatorname{Li}_ 2(z)+\frac{1}{2}\log^2(1-z)</math> |
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− | <math>z=e^{it}</math>, | + | <math>z=e^{it}</math>, <math>0 \leq t \leq \pi</math> 에서 |
위 식의 실수부를 취하면, 각각 다음 식을 얻는다. | 위 식의 실수부를 취하면, 각각 다음 식을 얻는다. | ||
60번째 줄: | 56번째 줄: | ||
<math>\sum_{n=1}^\infty \frac{H_n}{n+1}\sin (n+1)t=\frac{1}{2}(t-\pi)\log(2\sin\frac{t}{2})</math> | <math>\sum_{n=1}^\infty \frac{H_n}{n+1}\sin (n+1)t=\frac{1}{2}(t-\pi)\log(2\sin\frac{t}{2})</math> | ||
− | <math>\sum_{n=1}^\infty \frac{H_n}{n}\sin nt=\operatorname{Cl} | + | <math>\sum_{n=1}^\infty \frac{H_n}{n}\sin nt=\operatorname{Cl}_ 2(t)+\frac{1}{2}(t-\pi)\log(2\sin\frac{t}{2})</math> |
[[로바체프스키 함수|로바체프스키와 클라우센 함수]] | [[로바체프스키 함수|로바체프스키와 클라우센 함수]] | ||
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− | + | ==조화수열과 급수== | |
<math>\sum_{n=1}^{\infty}\frac{H_n^2}{(n+1)^2}=\frac{11\pi^4}{360}</math> | <math>\sum_{n=1}^{\infty}\frac{H_n^2}{(n+1)^2}=\frac{11\pi^4}{360}</math> | ||
80번째 줄: | 76번째 줄: | ||
<math>\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\frac{\pi^4}{72}</math> | <math>\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\frac{\pi^4}{72}</math> | ||
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− | + | ==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= | ||
* [[수학사연표 (역사)|수학사연표]] | * [[수학사연표 (역사)|수학사연표]] | ||
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− | + | ==메모== | |
http://sos440.tistory.com/202 | http://sos440.tistory.com/202 | ||
102번째 줄: | 98번째 줄: | ||
http://sos440.tistory.com/200 | http://sos440.tistory.com/200 | ||
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− | + | ==관련된 항목들== | |
* [[오일러상수, 감마]]<br> | * [[오일러상수, 감마]]<br> | ||
111번째 줄: | 107번째 줄: | ||
* [[로그 사인 적분 (log sine integrals)]]<br> | * [[로그 사인 적분 (log sine integrals)]]<br> | ||
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− | + | ==사전 형태의 자료== | |
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− | + | * http://ko.wikipedia.org/wiki/조화급수 | |
− | + | * http://en.wikipedia.org/wiki/Harmonic_series_(mathematics) | |
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* http://en.wikipedia.org/wiki/Harmonic_number | * http://en.wikipedia.org/wiki/Harmonic_number | ||
* http://mathworld.wolfram.com/HarmonicNumber.html | * http://mathworld.wolfram.com/HarmonicNumber.html | ||
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− | + | ==관련논문== | |
− | + | * [http://www.jstor.org/stable/2160718 On an Intriguing Integral and Some Series Related to \[Zeta](4)]<br> | |
− | + | ** David Borwein and Jonathan M. Borwein, Proceedings of the American Mathematical Society, Vol. 123, No. 4 (Apr., 1995), pp. 1191-1198 | |
− | * [http://www.jstor.org/stable/2160718 On an Intriguing Integral and Some Series Related to | ||
− | ** David Borwein and Jonathan M. Borwein, | ||
* 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/ | ||
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2012년 10월 7일 (일) 12:29 판
개요
\(\gamma=0.577215664901532860606512090\cdots\)
근사 공식==
- 오일러-맥클로린 공식 을 통해 다음을 얻는다
\(H_{n}=\sum_{k=1}^{n}\frac{1}{k}\sim \log n +\gamma+\frac{1}{2n}-\sum_{s=1}^{\infty}\frac{B_{2s}}{(2s)n^{2s}}\)
성질
\(H_{n-1}=H_n-\frac{1}{n}\)
\(H_ {n-1}^2=(H_n-\frac{1}{n})^2=H_n^2+\frac{1}{n^2}-\frac{2H_n}{n}\)
생성함수
\(\sum_{n=1}^\infty H_nz^n = \frac {-\ln(1-z)}{1-z}\)
생성함수의 응용
\(\sum_{n=1}^\infty \frac{H_n}{n+1}z^{n+1} =\frac{1}{2}\log^2(1-z)\)
\(\sum_{n=1}^\infty \frac{H_n}{n}z^n =\operatorname{Li}_ 2(z)+\frac{1}{2}\log^2(1-z)\)
\(z=e^{it}\), \(0 \leq t \leq \pi\) 에서
위 식의 실수부를 취하면, 각각 다음 식을 얻는다.
\(\sum_{n=1}^\infty \frac{H_n}{n+1}\sin (n+1)t=\frac{1}{2}(t-\pi)\log(2\sin\frac{t}{2})\)
\(\sum_{n=1}^\infty \frac{H_n}{n}\sin nt=\operatorname{Cl}_ 2(t)+\frac{1}{2}(t-\pi)\log(2\sin\frac{t}{2})\)
조화수열과 급수
\(\sum_{n=1}^{\infty}\frac{H_n^2}{(n+1)^2}=\frac{11\pi^4}{360}\)
\(\sum_{n=1}^{\infty}\frac{H_n^2}{n^2}=\frac{17\pi^4}{360}\)
\(\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\frac{\pi^4}{72}\)
역사
메모
관련된 항목들
- 오일러상수, 감마
- 조화급수와 조화 평균에서 '조화'란?
- 다이감마와 폴리감마 함수(digamma and polygamma functions)
- 로그 사인 적분 (log sine integrals)
사전 형태의 자료
- http://ko.wikipedia.org/wiki/조화급수
- http://en.wikipedia.org/wiki/Harmonic_series_(mathematics)
- http://en.wikipedia.org/wiki/Harmonic_number
- http://mathworld.wolfram.com/HarmonicNumber.html
관련논문
- On an Intriguing Integral and Some Series Related to \[Zeta(4)]
- David Borwein and Jonathan M. Borwein, Proceedings of the American Mathematical Society, Vol. 123, No. 4 (Apr., 1995), pp. 1191-1198
- http://www.jstor.org/action/doBasicSearch?Query=
- http://www.ams.org/mathscinet
- http://dx.doi.org/
\(H_{n}=\sum_{k=1}^{n}\frac{1}{k}\sim \log n +\gamma+\frac{1}{2n}-\sum_{s=1}^{\infty}\frac{B_{2s}}{(2s)n^{2s}}\)
- David Borwein and Jonathan M. Borwein, Proceedings of the American Mathematical Society, Vol. 123, No. 4 (Apr., 1995), pp. 1191-1198