"라마누잔의 정적분"의 두 판 사이의 차이
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| 1번째 줄: | 1번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">이 항목의 스프링노트 원문주소</h5> |
* [[라마누잔의 정적분]]<br> | * [[라마누잔의 정적분]]<br> | ||
| 7번째 줄: | 7번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">개요</h5> |
<math>\int_{0}^{\infty}\frac{x e^{-\sqrt{5}x}}{\cosh{x}}\,dx=\frac{1}{8}(\psi^{(1)}(\frac{1+\sqrt{5}}{4})-\psi^{(1)}(\frac{3+\sqrt{5}}{4}))</math> | <math>\int_{0}^{\infty}\frac{x e^{-\sqrt{5}x}}{\cosh{x}}\,dx=\frac{1}{8}(\psi^{(1)}(\frac{1+\sqrt{5}}{4})-\psi^{(1)}(\frac{3+\sqrt{5}}{4}))</math> | ||
| − | + | Integrate[(x Exp[-x Sqrt[5]])/Cosh[x], {x, 0, \[Infinity]}] // FullSimplify | |
| − | http://www.wolframalpha.com/input/?i=Integrate[(x+Exp[-x+Sqrt[5]])/Cosh[x],+{x,+0,+[Infinity]}]+ | + | [http://www.wolframalpha.com/input/?i=Integrate[%28x+Exp[-x+Sqrt[5]]%29/Cosh[x],+%7Bx,+0,+[Infinity]%7D]+ http://www.wolframalpha.com/input/?i=Integrate[(x+Exp[-x+Sqrt[5]])/Cosh[x],+{x,+0,+[Infinity]}]+] |
| + | |||
| + | [http://www.wolframalpha.com/input/?i=%28polygamma[1,%281%2Bsqrt%285%29%29/4]-polygamma[1,%283%2Bsqrt%285%29%29/4]%29/8 http://www.wolframalpha.com/input/?i=(polygamma[1,(1%2Bsqrt(5))/4]-polygamma[1,(3%2Bsqrt(5))/4])/8] | ||
| − | <math>\int_{0}^{\infty}\frac{x^{2}e^{-\sqrt{3}x}}{\sinh{x}}\,dx</math> | + | <math>\int_{0}^{\infty}\frac{x^{2}e^{-\sqrt{3}x}}{\sinh{x}}\,dx=-\frac{1}{4}\psi^{(2)}(\frac{1+\sqrt{3}}{4})</math> |
| + | |||
| + | Integrate[(x^2 Exp[-x Sqrt[3]])/Sinh[x], {x, 0, \[Infinity]}] //FullSimplify | ||
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| + | [http://www.wolframalpha.com/input/?i=integrate[%28x%5E2+Exp[-x+Sqrt[3]]%29/Sinh[x],+%7Bx,+0,+Infinity%7D] http://www.wolframalpha.com/input/?i=integrate[(x^2+Exp[-x+Sqrt[3]])/Sinh[x],+{x,+0,+Infinity}]] | ||
| + | |||
| + | [http://www.wolframalpha.com/input/?i=-polygamma[2,%281%2Bsqrt%283%29%29/2]/4 http://www.wolframalpha.com/input/?i=-polygamma[2,(1%2Bsqrt(3))/2]/4] | ||
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| 27번째 줄: | 37번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">재미있는 사실</h5> |
| + | |||
| + | * Math Overflow | ||
| + | * [http://mathoverflow.net/questions/66812/ramanujans-eccentric-integral-formula Ramanujan’s eccentric Integral formula] | ||
| − | + | 연분수 http://mathoverflow.net/questions/49866/applications-of-finite-continued-fractions/ | |
| − | * | + | * |
| + | * http://mathoverflow.net/search?q= | ||
* 네이버 지식인 http://kin.search.naver.com/search.naver?where=kin_qna&query= | * 네이버 지식인 http://kin.search.naver.com/search.naver?where=kin_qna&query= | ||
| 38번째 줄: | 52번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">역사</h5> |
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| − | <h5 style="line-height: 3.428em; margin | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">메모</h5> |
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| − | <h5 style="line-height: 3.428em; margin | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련된 항목들</h5> |
* [[트리감마 함수(trigamma function)]]<br> | * [[트리감마 함수(trigamma function)]]<br> | ||
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| − | <h5 style="line-height: 3.428em; margin | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">수학용어번역</h5> |
* 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 | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">사전 형태의 자료</h5> |
* http://ko.wikipedia.org/wiki/ | * http://ko.wikipedia.org/wiki/ | ||
| 84번째 줄: | 98번째 줄: | ||
* http://www.wolframalpha.com/input/?i= | * http://www.wolframalpha.com/input/?i= | ||
* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions] | * [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions] | ||
| − | * [http://www.research.att.com/ | + | * [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]<br> |
** http://www.research.att.com/~njas/sequences/?q= | ** http://www.research.att.com/~njas/sequences/?q= | ||
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| − | <h5 style="line-height: 3.428em; margin | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련논문</h5> |
* 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 | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련도서</h5> |
* 도서내검색<br> | * 도서내검색<br> | ||
| 115번째 줄: | 129번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련기사</h5> |
* 네이버 뉴스 검색 (키워드 수정)<br> | * 네이버 뉴스 검색 (키워드 수정)<br> | ||
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| − | <h5 style="line-height: 3.428em; margin | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">블로그</h5> |
* 구글 블로그 검색<br> | * 구글 블로그 검색<br> | ||
2011년 9월 17일 (토) 03:05 판
이 항목의 스프링노트 원문주소
개요
\(\int_{0}^{\infty}\frac{x e^{-\sqrt{5}x}}{\cosh{x}}\,dx=\frac{1}{8}(\psi^{(1)}(\frac{1+\sqrt{5}}{4})-\psi^{(1)}(\frac{3+\sqrt{5}}{4}))\)
Integrate[(x Exp[-x Sqrt[5]])/Cosh[x], {x, 0, \[Infinity]}] // FullSimplify
[%28x+Exp[-x+Sqrt[5]%29/Cosh[x],+%7Bx,+0,+[Infinity]%7D]+ http://www.wolframalpha.com/input/?i=Integrate[(x+Exp[-x+Sqrt[5]])/Cosh[x],+{x,+0,+[Infinity]}]+]
[1,%281%2Bsqrt%285%29%29/4-polygamma[1,%283%2Bsqrt%285%29%29/4]%29/8 http://www.wolframalpha.com/input/?i=(polygamma[1,(1%2Bsqrt(5))/4]-polygamma[1,(3%2Bsqrt(5))/4])/8]
\(\int_{0}^{\infty}\frac{x^{2}e^{-\sqrt{3}x}}{\sinh{x}}\,dx=-\frac{1}{4}\psi^{(2)}(\frac{1+\sqrt{3}}{4})\)
Integrate[(x^2 Exp[-x Sqrt[3]])/Sinh[x], {x, 0, \[Infinity]}] //FullSimplify
[%28x%5E2+Exp[-x+Sqrt[3]%29/Sinh[x],+%7Bx,+0,+Infinity%7D] http://www.wolframalpha.com/input/?i=integrate[(x^2+Exp[-x+Sqrt[3]])/Sinh[x],+{x,+0,+Infinity}]]
[2,%281%2Bsqrt%283%29%29/2/4 http://www.wolframalpha.com/input/?i=-polygamma[2,(1%2Bsqrt(3))/2]/4]
Berndt, B. C. and Rankin, R. A. Ramanujan: Letters and Commentary. Providence, RI: Amer. Math. Soc., 1995.
재미있는 사실
- Math Overflow
- Ramanujan’s eccentric Integral formula
연분수 http://mathoverflow.net/questions/49866/applications-of-finite-continued-fractions/
- http://mathoverflow.net/search?q=
- 네이버 지식인 http://kin.search.naver.com/search.naver?where=kin_qna&query=
역사
메모
관련된 항목들
수학용어번역
- 단어사전 http://www.google.com/dictionary?langpair=en%7Cko&q=
- 발음사전 http://www.forvo.com/search/
- 대한수학회 수학 학술 용어집
- 남·북한수학용어비교
- 대한수학회 수학용어한글화 게시판
사전 형태의 자료
- http://ko.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/
- http://mathworld.wolfram.com/RamanujanContinuedFractions.html
- http://www.wolframalpha.com/input/?i=
- NIST Digital Library of Mathematical Functions
- The On-Line Encyclopedia of Integer Sequences
관련논문
관련도서
- 도서내검색
- 도서검색
관련기사
- 네이버 뉴스 검색 (키워드 수정)