"트리감마 함수(trigamma function)"의 두 판 사이의 차이
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개요==
성질==
\(\psi^{(1)}(z + 1) - \psi^{(1)}(z) =-\frac{1}{z^{2}}\)
후르비츠 제타함수==
클라우센 함수와의 관계==
Pythagoras0 (토론 | 기여) 잔글 (찾아 바꾸기 – “</h5>” 문자열을 “==” 문자열로) |
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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%;">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== |
* [[트리감마 함수(trigamma function)]]<br> | * [[트리감마 함수(trigamma function)]]<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%;">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== |
* [[다이감마 함수(digamma function)]] 함수의 도함수<br> | * [[다이감마 함수(digamma function)]] 함수의 도함수<br> | ||
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| − | <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">성질 | + | <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">성질== |
<math>\psi^{(1)}(z + 1) - \psi^{(1)}(z) =-\frac{1}{z^{2}}</math> | <math>\psi^{(1)}(z + 1) - \psi^{(1)}(z) =-\frac{1}{z^{2}}</math> | ||
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| − | <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">덧셈공식 | + | <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">덧셈공식== |
<math>\psi^{(1)}(z)+ \psi^{(1)}\left(z + \frac{1}{m}\right) + \cdots+ \psi^{(1)}\left(z + \frac{m-1}{m}\right) = m^{2}\psi^{(1)}(mz)</math> | <math>\psi^{(1)}(z)+ \psi^{(1)}\left(z + \frac{1}{m}\right) + \cdots+ \psi^{(1)}\left(z + \frac{m-1}{m}\right) = m^{2}\psi^{(1)}(mz)</math> | ||
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| − | <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">후르비츠 제타함수 | + | <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">후르비츠 제타함수== |
* [[후르비츠 제타함수(Hurwitz zeta function)]]<br><math>\psi'(z)=\zeta(2,z)=\sum_{n=0}^\infty \frac{1}{(n+z)^2}</math><br> | * [[후르비츠 제타함수(Hurwitz zeta function)]]<br><math>\psi'(z)=\zeta(2,z)=\sum_{n=0}^\infty \frac{1}{(n+z)^2}</math><br> | ||
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| − | <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">클라우센 함수와의 관계 | + | <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">클라우센 함수와의 관계== |
| − | * [[클라우센 함수(Clausen function)]]<br>[[클라우센 함수(Clausen function)|]]<math>\operatorname{Cl}_2(\frac{\pi}{3})=\frac{\sqrt{3}}{12}(\psi^{(1)}(\frac{1}{3})-\psi^{(1)}(\frac{2}{3}))</math><br> 여기서 <math>\psi^{(1)}</math>는 트리감마(trigamma)함수.<br> | + | * [[클라우센 함수(Clausen function)]]<br>[[클라우센 함수(Clausen function)|클라우센 함수]]<math>\operatorname{Cl}_2(\frac{\pi}{3})=\frac{\sqrt{3}}{12}(\psi^{(1)}(\frac{1}{3})-\psi^{(1)}(\frac{2}{3}))</math><br> 여기서 <math>\psi^{(1)}</math>는 트리감마(trigamma)함수.<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%;">재미있는 사실 | + | <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%;">재미있는 사실== |
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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%;">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== |
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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%;">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== |
http://www.wolframalpha.com/input/?i=(polylog[2,exp(-i*2pi/3)]-polylog[2,exp(i*2pi/3)])*i/2 | http://www.wolframalpha.com/input/?i=(polylog[2,exp(-i*2pi/3)]-polylog[2,exp(i*2pi/3)])*i/2 | ||
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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%;">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== |
* [[로바체프스키 함수|로바체프스키와 클라우센 함수]]<br> | * [[로바체프스키 함수|로바체프스키와 클라우센 함수]]<br> | ||
| 107번째 줄: | 107번째 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== |
* 단어사전 http://www.google.com/dictionary?langpair=en|ko&q= | * 단어사전 http://www.google.com/dictionary?langpair=en|ko&q= | ||
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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%;">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== |
* http://ko.wikipedia.org/wiki/ | * http://ko.wikipedia.org/wiki/ | ||
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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= | ||
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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%;">관련도서 | + | <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== |
* 도서내검색<br> | * 도서내검색<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%;">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> | ||
| 169번째 줄: | 169번째 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) 13:12 판
이 항목의 스프링노트 원문주소==
개요==
- 다이감마 함수(digamma function) 함수의 도함수
- 다음과 같이 주어진다
\(\psi'(z)=\sum_{n=0}^\infty \frac{1}{(n+z)^2}\)
\(\psi'(z)=\sum_{n=0}^\infty \frac{1}{(n+z)^2}\)
성질==
\(\psi^{(1)}(z + 1) - \psi^{(1)}(z) =-\frac{1}{z^{2}}\)
- 차분방정식에의 응용
덧셈공식== \(\psi^{(1)}(z)+ \psi^{(1)}\left(z + \frac{1}{m}\right) + \cdots+ \psi^{(1)}\left(z + \frac{m-1}{m}\right) = m^{2}\psi^{(1)}(mz)\)
후르비츠 제타함수==
- 후르비츠 제타함수(Hurwitz zeta function)
\(\psi'(z)=\zeta(2,z)=\sum_{n=0}^\infty \frac{1}{(n+z)^2}\)
\(\psi'(z)=\zeta(2,z)=\sum_{n=0}^\infty \frac{1}{(n+z)^2}\)
클라우센 함수와의 관계==
- 클라우센 함수(Clausen function)
클라우센 함수\(\operatorname{Cl}_2(\frac{\pi}{3})=\frac{\sqrt{3}}{12}(\psi^{(1)}(\frac{1}{3})-\psi^{(1)}(\frac{2}{3}))\)
여기서 \(\psi^{(1)}\)는 트리감마(trigamma)함수.
http://mathworld.wolfram.com/GiesekingsConstant.html
http://www.research.att.com/~njas/sequences/A143298
http://www.wolframalpha.com/input/?i=Gieseking's+constant.
http://www.wolframalpha.com/input/?i=sqrt(3)*(trigamma(1/3)-trigamma(2/3))/12
클라우센 함수\(\operatorname{Cl}_2(\frac{\pi}{3})=\frac{\sqrt{3}}{12}(\psi^{(1)}(\frac{1}{3})-\psi^{(1)}(\frac{2}{3}))\)
여기서 \(\psi^{(1)}\)는 트리감마(trigamma)함수.