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Pythagoras0 (토론 | 기여) 잔글 (찾아 바꾸기 – “==관련도서== * 도서내검색<br> ** http://books.google.com/books?q= ** http://book.daum.net/search/contentSearch.do?query= * 도서검색<br> ** http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-key) |
Pythagoras0 (토론 | 기여) |
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==개요== | ==개요== | ||
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* 수열의 합과 적분을 연결해주는 공식 | * 수열의 합과 적분을 연결해주는 공식 | ||
| − | + | :<math>\sum _{i=a}^{b-1} f(i)=\int_a^b f(x) \, dx+\frac{1}{2} (f(a)-f(b))+\frac{1}{12} \left(f'(b)-f'(a)\right)+\frac{1}{720} \left(f^{(3)}(a)-f^{(3)}(b)\right)+\frac{f^{(5)}(b)-f^{(5)}(a)}{30240}+\frac{f^{(7)}(a)-f^{(7)}(b)}{1209600}+\cdots</math> | |
| − | <math>\sum _{i=a}^{b-1} f(i)=\int_a^b f(x) \, dx+\frac{1}{2} (f(a)-f(b))+\frac{1}{12} \left(f'(b)-f'(a)\right)+\frac{1}{720} \left(f^{(3)}(a)-f^{(3)}(b)\right)+\frac{f^{(5)}(b)-f^{(5)}(a)}{30240}+\frac{f^{(7)}(a)-f^{(7)}(b)}{1209600}+\cdots</math> | ||
* 오차항 | * 오차항 | ||
| − | + | :<math>\sum_{i=a}^{b-1} f(i) = \int^b_a f(x)\,dx+\sum_{k=1}^p\frac{B_k}{k!}\left(f^{(k-1)}(b)-f^{(k-1)}(a)\right)+R</math> | |
| − | <math>\sum_{i=a}^{b-1} f(i) = \int^b_a f(x)\,dx+\sum_{k=1}^p\frac{B_k}{k!}\left(f^{(k-1)}(b)-f^{(k-1)}(a)\right)+R</math> | ||
여기서 | 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> | ||
<math>B_0=1</math>, <math>B_1=-{1 \over 2}</math>, <math>B_2={1\over 6}</math>, <math>B_3=0</math>, <math>B_4=-\frac{1}{30}</math>, <math>B_5=0</math>, <math>B_6=\frac{1}{42}</math>, <math>B_8=-\frac{1}{30}</math>, <math>B_{10}=\frac{5}{66}</math>, <math>B_{12}=-\frac{691}{2730}</math>,<math>B_{14}=\frac{7}{6}</math> 는 [[베르누이 수]] | <math>B_0=1</math>, <math>B_1=-{1 \over 2}</math>, <math>B_2={1\over 6}</math>, <math>B_3=0</math>, <math>B_4=-\frac{1}{30}</math>, <math>B_5=0</math>, <math>B_6=\frac{1}{42}</math>, <math>B_8=-\frac{1}{30}</math>, <math>B_{10}=\frac{5}{66}</math>, <math>B_{12}=-\frac{691}{2730}</math>,<math>B_{14}=\frac{7}{6}</math> 는 [[베르누이 수]] | ||
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* https://docs.google.com/leaf?id=0B8XXo8Tve1cxN2U5NmI1Y2YtNjYyMi00OWEwLWI3MGQtNTRmYjdiYWM4ZTM3&sort=name&layout=list&num=50 | * https://docs.google.com/leaf?id=0B8XXo8Tve1cxN2U5NmI1Y2YtNjYyMi00OWEwLWI3MGQtNTRmYjdiYWM4ZTM3&sort=name&layout=list&num=50 | ||
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==사전자료== | ==사전자료== | ||
| + | * http://en.wikipedia.org/wiki/Euler's_summation_formula | ||
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==관련논문== | ==관련논문== | ||
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* [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/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 | * [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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2014년 10월 27일 (월) 08:48 판
개요
- 수열의 합과 적분을 연결해주는 공식
\[\sum _{i=a}^{b-1} f(i)=\int_a^b f(x) \, dx+\frac{1}{2} (f(a)-f(b))+\frac{1}{12} \left(f'(b)-f'(a)\right)+\frac{1}{720} \left(f^{(3)}(a)-f^{(3)}(b)\right)+\frac{f^{(5)}(b)-f^{(5)}(a)}{30240}+\frac{f^{(7)}(a)-f^{(7)}(b)}{1209600}+\cdots\]
- 오차항
\[\sum_{i=a}^{b-1} f(i) = \int^b_a f(x)\,dx+\sum_{k=1}^p\frac{B_k}{k!}\left(f^{(k-1)}(b)-f^{(k-1)}(a)\right)+R\]
여기서 \[\left|R\right|\leq\frac{2}{(2\pi)^{2(p+1)}}\int_0^n\left|f^{(p)}(x)\right|\,dx\]
\(B_0=1\), \(B_1=-{1 \over 2}\), \(B_2={1\over 6}\), \(B_3=0\), \(B_4=-\frac{1}{30}\), \(B_5=0\), \(B_6=\frac{1}{42}\), \(B_8=-\frac{1}{30}\), \(B_{10}=\frac{5}{66}\), \(B_{12}=-\frac{691}{2730}\),\(B_{14}=\frac{7}{6}\) 는 베르누이 수
\(\frac{B_k}{k!}\) 는 \(\{1, -1/2, 1/12, 0, -1/720, 0, 1/30240, 0, -1/1209600, 0, 1/47900160, 0, -691/1307674368000, 0, 1/74724249600\}\)
응용1.
응용2.
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
매스매티카 파일 및 계산 리소스
사전자료
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