"오일러-맥클로린 공식"의 두 판 사이의 차이
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| 7번째 줄: | 7번째 줄: | ||
| − | <h5> | + | <h5>개요</h5> |
* 수열의 합과 적분을 연결해주는 공식 | * 수열의 합과 적분을 연결해주는 공식 | ||
| − | + | <math>\sum_{i=0}^{n-1} f(i) = \int^n_0f(x)\,dx+\sum_{k=1}^p\frac{B_k}{k!}\left(f^{(k-1)}(n)-f^{(k-1)}(0)\right)+R</math> | |
| − | + | 여기서 | |
<math>\left|R\right|\leq\frac{2}{(2\pi)^{2(p+1)}}\int_0^n\left|f^{(p)}(x)\right|\,dx</math> | <math>\left|R\right|\leq\frac{2}{(2\pi)^{2(p+1)}}\int_0^n\left|f^{(p)}(x)\right|\,dx</math> | ||
| 20번째 줄: | 20번째 줄: | ||
<math>\frac{B_k}{k!}</math> 는 <math>\{1, -1/2, 1/12, 0, -1/720, 0, 1/30240, 0, -1/1209600, 0, 1/47900160, 0, -691/1307674368000, 0, 1/74724249600\}</math> | <math>\frac{B_k}{k!}</math> 는 <math>\{1, -1/2, 1/12, 0, -1/720, 0, 1/30240, 0, -1/1209600, 0, 1/47900160, 0, -691/1307674368000, 0, 1/74724249600\}</math> | ||
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| + | <math>\sum_{i=0}^{n-1} f(i) = \int^n_0f(x)\,dx-\frac{1}{2}(f(n)-f(0))+\frac{1}{12}(f'(n)-f'(0))-\frac{1}{720}(f^{(3)}(n)-f^{(3)}(0))+\frac{1}{30240}(f^{(5)}(n)-f^{(5)}(0))-\frac{1}{1209600}(f^{(7)}(n)-f^{(7)}(0))+\cdots</math> | ||
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| 45번째 줄: | 47번째 줄: | ||
* [[오일러상수, 감마]] | * [[오일러상수, 감마]] | ||
* [[ζ(2)의 계산, 오일러와 바젤문제(완전제곱수의 역수들의 합)|오일러와 바젤문제(완전제곱수의 역수들의 합)]] | * [[ζ(2)의 계산, 오일러와 바젤문제(완전제곱수의 역수들의 합)|오일러와 바젤문제(완전제곱수의 역수들의 합)]] | ||
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| 100번째 줄: | 98번째 줄: | ||
<h5>관련논문</h5> | <h5>관련논문</h5> | ||
| − | * | + | * Euler-Maclaurin summation formula ([[2637804/attachments/1168462|pdf]]) , E. Hairer (Author), G. Wanner, From [http://www.amazon.com/Analysis-History-Undergraduate-Mathematics-Readings/dp/0387945512 Analysis by Its History], 160-169p |
| − | + | * [http://www.math.nmsu.edu/%7Edavidp/euler2k2.pdf Dances between continuous and discrete: Euler's summation formula] ,David J. Pengelley, in: Robert Bradley and Ed Sandifer (Eds), Proceedings, Euler 2K+2 Conference (Rumford, Maine, 2002) , Euler Society, 2003. | |
| − | + | * [http://dx.doi.org/10.2307%2F2589145 An Elementary View of Euler's Summation Formula], Tom M. Apostol, <cite>[http://www.jstor.org/action/showPublication?journalCode=amermathmont The American Mathematical Monthly]</cite>, Vol. 106, No. 5 (May, 1999), pp. 409-418 | |
| − | * [http://www.math.nmsu.edu/%7Edavidp/euler2k2.pdf Dances between continuous and discrete: Euler's summation formula] | + | * [http://www.jstor.org/stable/2690625 The Euler-Maclaurin and Taylor Formulas: Twin, Elementary Derivations] , Vito Lampret, <cite>Mathematics Magazine</cite>, Vol. 74, No. 2 (Apr., 2001), pp. 109-122 |
| − | + | * [http://www.jstor.org/stable/2301097 An Euler Summation Formula] , Irwin Roman, <cite>The American Mathematical Monthly</cite>, Vol. 43, No. 1 (Jan., 1936), pp. 9-21 | |
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| − | * [http://dx.doi.org/10.2307%2F2589145 An Elementary View of Euler's Summation Formula] | ||
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| − | * [http://www.jstor.org/stable/2690625 The Euler-Maclaurin and Taylor Formulas: Twin, Elementary Derivations] | ||
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| − | * [http://www.jstor.org/stable/2301097 An Euler Summation Formula] | ||
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2011년 5월 1일 (일) 08:30 판
이 항목의 스프링노트 원문주소
개요
- 수열의 합과 적분을 연결해주는 공식
\(\sum_{i=0}^{n-1} f(i) = \int^n_0f(x)\,dx+\sum_{k=1}^p\frac{B_k}{k!}\left(f^{(k-1)}(n)-f^{(k-1)}(0)\right)+R\)
여기서
\(\left|R\right|\leq\frac{2}{(2\pi)^{2(p+1)}}\int_0^n\left|f^{(p)}(x)\right|\,dx\)
\(B_0=1\), \(B_1=-{1 \over 2}\), \(B_2={1\over 6}\), \(B_3=0\), \(B_4=-\frac{1}{30}\), \(B_5=0\), \(B_6=\frac{1}{42}\), \(B_8=-\frac{1}{30}\), \(B_{10}=\frac{5}{66}\), \(B_{12}=-\frac{691}{2730}\),\(B_{14}=\frac{7}{6}\)
\(\frac{B_k}{k!}\) 는 \(\{1, -1/2, 1/12, 0, -1/720, 0, 1/30240, 0, -1/1209600, 0, 1/47900160, 0, -691/1307674368000, 0, 1/74724249600\}\)
\(\sum_{i=0}^{n-1} f(i) = \int^n_0f(x)\,dx-\frac{1}{2}(f(n)-f(0))+\frac{1}{12}(f'(n)-f'(0))-\frac{1}{720}(f^{(3)}(n)-f^{(3)}(0))+\frac{1}{30240}(f^{(5)}(n)-f^{(5)}(0))-\frac{1}{1209600}(f^{(7)}(n)-f^{(7)}(0))+\cdots\)
유용한 표현
\(\sum_{i=0}^{n-1} f(i) = \sum_{k=0}^p\frac{B_k}{k!}\left(f^{(k-1)}(n)-f^{(k-1)}(0)\right)+R\)
단, \(f^{(-1)}(x)=\int f(x)\,dx\) 라고 쓰자.
응용
재미있는 사실
- 오일러의 계산에 중요하게 활용되었다
관련된 고교수학 또는 대학수학
관련된 항목들
사전자료
관련도서
- 도서내검색
- 도서검색
관련논문
- Euler-Maclaurin summation formula (pdf) , E. Hairer (Author), G. Wanner, From Analysis by Its History, 160-169p
- Dances between continuous and discrete: Euler's summation formula ,David J. Pengelley, in: Robert Bradley and Ed Sandifer (Eds), Proceedings, Euler 2K+2 Conference (Rumford, Maine, 2002) , Euler Society, 2003.
- An Elementary View of Euler's Summation Formula, Tom M. Apostol, The American Mathematical Monthly, Vol. 106, No. 5 (May, 1999), pp. 409-418
- The Euler-Maclaurin and Taylor Formulas: Twin, Elementary Derivations , Vito Lampret, Mathematics Magazine, Vol. 74, No. 2 (Apr., 2001), pp. 109-122
- An Euler Summation Formula , Irwin Roman, The American Mathematical Monthly, Vol. 43, No. 1 (Jan., 1936), pp. 9-21
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