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개요==
정의==
예1==
예2==
Pythagoras0 (토론 | 기여) 잔글 (찾아 바꾸기 – “</h5>” 문자열을 “==” 문자열로) |
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| 1번째 줄: | 1번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">이 항목의 스프링노트 원문주소 | + | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">이 항목의 스프링노트 원문주소== |
* [[푸리에 급수]]<br> | * [[푸리에 급수]]<br> | ||
| 7번째 줄: | 7번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">개요 | + | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">개요== |
* 주어진 함수의 삼각함수를 이용한 급수표현<br> | * 주어진 함수의 삼각함수를 이용한 급수표현<br> | ||
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| − | <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">정의 | + | <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">정의== |
* <math>2\pi</math>를 주기로 가지는 함수 <math>f</math><br> | * <math>2\pi</math>를 주기로 가지는 함수 <math>f</math><br> | ||
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| − | <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">예1 | + | <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">예1== |
* <math>f(x)=x</math>, <math>-\pi < x < \pi</math><br><math>f(x)\sim2\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n} \sin(nx)</math><br> | * <math>f(x)=x</math>, <math>-\pi < x < \pi</math><br><math>f(x)\sim2\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n} \sin(nx)</math><br> | ||
| 42번째 줄: | 42번째 줄: | ||
| − | <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">예2 | + | <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">예2== |
* [[로바체프스키 함수|로바체프스키와 클라우센 함수]]<br><math>0 \leq \theta \leq 2\pi</math> 일때,<math>Cl_2(\theta)=\sum_{n=1}^\infty \frac{1}{n^2}\sin n\theta</math><br> | * [[로바체프스키 함수|로바체프스키와 클라우센 함수]]<br><math>0 \leq \theta \leq 2\pi</math> 일때,<math>Cl_2(\theta)=\sum_{n=1}^\infty \frac{1}{n^2}\sin n\theta</math><br> | ||
| 52번째 줄: | 52번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">역사 | + | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">역사== |
* [[수학사연표 (역사)|수학사연표]] | * [[수학사연표 (역사)|수학사연표]] | ||
| 60번째 줄: | 60번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">메모 | + | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">메모== |
<math>\hat{f}(n) = \frac{1}{2\pi}\int_{-\pi}^\pi f(x) e^{inx}\, dx</math> | <math>\hat{f}(n) = \frac{1}{2\pi}\int_{-\pi}^\pi f(x) e^{inx}\, dx</math> | ||
| 70번째 줄: | 70번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">관련된 항목들 | + | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">관련된 항목들== |
* [[편미분방정식]]<br> | * [[편미분방정식]]<br> | ||
| 79번째 줄: | 79번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">수학용어번역 | + | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">수학용어번역== |
* http://www.google.com/dictionary?langpair=en|ko&q= | * http://www.google.com/dictionary?langpair=en|ko&q= | ||
| 90번째 줄: | 90번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">사전 형태의 자료 | + | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">사전 형태의 자료== |
* <br> | * <br> | ||
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| − | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">관련논문 | + | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">관련논문== |
* http://www.jstor.org/action/doBasicSearch?Query= | * http://www.jstor.org/action/doBasicSearch?Query= | ||
| 112번째 줄: | 112번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">관련도서 및 추천도서 | + | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">관련도서 및 추천도서== |
* 도서내검색<br> | * 도서내검색<br> | ||
| 126번째 줄: | 126번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">관련기사 | + | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">관련기사== |
* 네이버 뉴스 검색 (키워드 수정)<br> | * 네이버 뉴스 검색 (키워드 수정)<br> | ||
| 137번째 줄: | 137번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">블로그 | + | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">블로그== |
* 구글 블로그 검색<br> | * 구글 블로그 검색<br> | ||
2012년 11월 1일 (목) 13:14 판
이 항목의 스프링노트 원문주소==
개요==
- 주어진 함수의 삼각함수를 이용한 급수표현
- 열방정식을 푸는 과정에서 푸리에가 발견
정의==
- \(2\pi\)를 주기로 가지는 함수 \(f\)
- 푸리에 계수의 정의
\(a_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x) \cos(nx)\, dx, \quad n \ge 0\)
\(b_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x) \sin(nx)\, dx, \quad n \ge 1\)
- 푸리에 급수
\(f(x)\sim \frac{a_0}{2} + \sum_{n=1}^\infty \, [a_n \cos(nx) + b_n \sin(nx)]\)
\(a_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x) \cos(nx)\, dx, \quad n \ge 0\)
\(b_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x) \sin(nx)\, dx, \quad n \ge 1\)
\(f(x)\sim \frac{a_0}{2} + \sum_{n=1}^\infty \, [a_n \cos(nx) + b_n \sin(nx)]\)
예1==
- \(f(x)=x\), \(-\pi < x < \pi\)
\(f(x)\sim2\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n} \sin(nx)\)
- \(f(x)=\frac{\pi-x}{2}\),\(0 < x \leq \pi\)
\(f(x) \sim \sum_{n=1}^{\infty}\frac{1}{n}\sin n x\)
- \(f(x)=x^2\), \(-\pi < x < \pi\)
\(f(x)\sim \frac{\pi^2}{3}+4\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^2} \cos(nx)\)
\(f(x)\sim2\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n} \sin(nx)\)
\(f(x) \sim \sum_{n=1}^{\infty}\frac{1}{n}\sin n x\)
\(f(x)\sim \frac{\pi^2}{3}+4\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^2} \cos(nx)\)
예2==
- 로바체프스키와 클라우센 함수
\(0 \leq \theta \leq 2\pi\) 일때,\(Cl_2(\theta)=\sum_{n=1}^\infty \frac{1}{n^2}\sin n\theta\)
- 로그감마 함수
\(\log\Gamma(x)=\log\sqrt{2\pi}-\frac{1}{2}\log(2\sin\pi x)+\frac{1}{2}(\gamma+2\log\sqrt{2\pi})(1-2x)+\frac{1}{\pi}\sum_{n=1}^{\infty}\frac{\log n}{n}\sin 2\pi nx\)
\(0 \leq \theta \leq 2\pi\) 일때,\(Cl_2(\theta)=\sum_{n=1}^\infty \frac{1}{n^2}\sin n\theta\)
\(\log\Gamma(x)=\log\sqrt{2\pi}-\frac{1}{2}\log(2\sin\pi x)+\frac{1}{2}(\gamma+2\log\sqrt{2\pi})(1-2x)+\frac{1}{\pi}\sum_{n=1}^{\infty}\frac{\log n}{n}\sin 2\pi nx\)