"로그감마 함수"의 두 판 사이의 차이

수학노트
둘러보기로 가기 검색하러 가기
 
(사용자 2명의 중간 판 22개는 보이지 않습니다)
1번째 줄: 1번째 줄:
<h5 style="margin: 0px; line-height: 3.428em; 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>
+
==개요==
  
* [[로그감마 함수]]
+
  
 
+
  
 
+
==후르비츠 제타함수==
  
<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">개요</h5>
+
*  Lerch의 공식 : [[후르비츠 제타함수(Hurwitz zeta function)]]의 미분:<math>\frac{\partial }{\partial s}\zeta(s,a)|_{s=0} =\log \frac{\Gamma(a)}{\sqrt{2\pi}}</math>
  
 
+
  
 
+
  
<h5>후르비츠 제타함수</h5>
+
==적분표현==
  
Lerch의 공식 : [[후르비츠 제타함수(Hurwitz zeta function)]]의 미분<br><math>\frac{\partial }{\partial s}\zeta(s,a)|_{s=0} =\log \frac{\Gamma(a)}{\sqrt{2\pi}}</math><br>
+
Binet's second expression:<math>\operatorname{Re} z > 0 </math> 일 때, <math>\log \Gamma(z)=(z-\frac{1}{2})\log z -z+\frac{1}{2}\log 2\pi+ 2\int_0^{\infty}\frac{\tan^{-1}(t/z)}{e^{2\pi t} -1}dt</math>http://dlmf.nist.gov/5/9/ 참고
  
 
+
  
 
+
  
<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">적분표현</h5>
+
==쿰머의 푸리에 급수==
  
Binet's second expression<br><math>\operatorname{Re} z > 0 </math> 일 때, <math>\log \Gamma(z)=(z-\frac{1}{2})\log z -z+\frac{1}{2}\log 2\pi+ 2\int_0^{\infty}\frac{\tan^{-1}(t/z)}{e^{2\pi t} -1}dt</math><br>http://dlmf.nist.gov/5/9/ 참고<br>
+
쿰머 (1847):<math>\begin{eqnarray}\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_{k=1}^{\infty}\frac{\log k}{k}\sin 2\pi kx \nonumber \\ =(\frac{1}{2}-x)(\gamma+\log 2)+(1-x)\log \pi -\frac{1}{2}\log(\sin\pi x)+\frac{1}{\pi}\sum_{k=1}^{\infty}\frac{\log k}{k}\sin 2\pi kx \nonumber  \end{eqnarray} </math>
  
 
+
  
 
+
  
<h5>쿰머의 푸리에 급수</h5>
+
==테일러 급수==
  
* 쿰머 (1847)<br><math>\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_{k=1}^{\infty}\frac{\log k}{k}\sin 2\pi kx</math><br>http://arxiv.org/ftp/arxiv/papers/0903/0903.4323.pdf<br>
+
* [[로그감마 함수]]의 테일러 급수 (http://www.wolframalpha.com/input/?i=taylor+series+of+log+gamma(1%2Bx)+at+x%3D0):<math>\log\Gamma(1+x) =-\gamma x+\sum_{k=2}^{\infty}(-1)^k \frac{\zeta(k)}{k}x^k</math>
 +
* [[정수에서의 리만제타함수의 값]]
  
 
+
  
* [http://www.math.titech.ac.jp/%7Etosho/Preprints/pdf/128.pdf http://www.math.titech.ac.jp/~tosho/Preprints/pdf/128.pdf]
+
* [http://www.math.tulane.edu/%7Evhm/papers_html/log-gamma.pdf http://www.math.tulane.edu/~vhm/papers_html/log-gamma.pdf]
 
*  
 
  
 
+
==정적분==
 
 
 
 
 
 
<h5>정적분</h5>
 
  
 
<math>\int_{0}^{1}\log\Gamma(x)\,dx=\log\sqrt{2\pi}</math>
 
<math>\int_{0}^{1}\log\Gamma(x)\,dx=\log\sqrt{2\pi}</math>
  
 
+
 
 
 
 
  
 
+
<math>\int_{0}^{\frac{1}{2}}\log\Gamma(x+1)\,dx=-\frac{1}{2}-\frac{7}{24}\log 2+\frac{1}{4}\log \pi+\frac{3}{2}\log A</math>
  
 
+
A는 [[Glaisher–Kinkelin 상수]]
  
<h5>재미있는 사실</h5>
+
  
 
+
  
* Math Overflow http://mathoverflow.net/search?q=
+
==스털링 공식==
* 네이버 지식인 http://kin.search.naver.com/search.naver?where=kin_qna&query=
 
  
 
+
* [[스털링 공식]]
  
 
+
  
<h5>역사</h5>
+
  
 
 
  
* http://www.google.com/search?hl=en&tbs=tl:1&q=
+
* [[수학사연표 (역사)|수학사연표]]
 
*  
 
  
 
+
==메모==
  
 
+
  
<h5>메모</h5>
+
  
 
+
==관련된 항목들==
 
 
 
 
 
 
<h5>관련된 항목들</h5>
 
  
 
* [[후르비츠 제타함수(Hurwitz zeta function)]]
 
* [[후르비츠 제타함수(Hurwitz zeta function)]]
  
 
+
  
 
+
 +
  
<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">수학용어번역</h5>
+
==사전 형태의 자료==
 
 
* 단어사전 http://www.google.com/dictionary?langpair=en|ko&q=
 
* 발음사전 http://www.forvo.com/search/
 
* [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집]<br>
 
** http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
 
* [http://www.nktech.net/science/term/term_l.jsp?l_mode=cate&s_code_cd=MA 남·북한수학용어비교]
 
* [http://kms.or.kr/home/kor/board/bulletin_list_subject.asp?bulletinid=%7BD6048897-56F9-43D7-8BB6-50B362D1243A%7D&boardname=%BC%F6%C7%D0%BF%EB%BE%EE%C5%E4%B7%D0%B9%E6&globalmenu=7&localmenu=4 대한수학회 수학용어한글화 게시판]
 
 
 
 
 
 
 
 
 
 
 
<h5>사전 형태의 자료</h5>
 
 
 
* http://ko.wikipedia.org/wiki/
 
* http://en.wikipedia.org/wiki/
 
 
* http://mathworld.wolfram.com/LogGammaFunction.html
 
* http://mathworld.wolfram.com/LogGammaFunction.html
 
* http://www.wolframalpha.com/input/?i=Loggamma
 
* http://www.wolframalpha.com/input/?i=Loggamma
* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
 
* [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=
 
 
 
 
 
 
 
 
<h5>관련논문</h5>
 
 
* [http://www.math.tulane.edu/~vhm/papers_html/log-gamma.pdf INTEGRALS OF POWERS OF LOGGAMMA]<br>
 
** TEWODROS AMDEBERHAN, MARK W. COFFEY, OLIVIER ESPINOSA, CHRISTOPH KOUTSCHAN, DANTE V. MANNA, AND VICTOR H. MOLL
 
 
* http://www.jstor.org/action/doBasicSearch?Query=
 
* http://www.ams.org/mathscinet
 
* http://dx.doi.org/
 
 
 
 
 
 
 
 
<h5>관련도서</h5>
 
 
*  도서내검색<br>
 
** http://books.google.com/books?q=
 
** http://book.daum.net/search/contentSearch.do?query=
 
*  도서검색<br>
 
** http://books.google.com/books?q=
 
** http://book.daum.net/search/mainSearch.do?query=
 
** http://book.daum.net/search/mainSearch.do?query=
 
 
 
 
 
 
 
 
<h5>관련기사</h5>
 
 
*  네이버 뉴스 검색 (키워드 수정)<br>
 
** http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
 
** http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
 
** http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
 
 
 
 
 
 
 
  
<h5>블로그</h5>
+
  
*  구글 블로그 검색<br>
+
==관련논문==
** http://blogsearch.google.com/blogsearch?q=
+
* Diamond, Harold G., and Armin Straub. “Bounds for the Logarithm of the Euler Gamma Function and Its Derivatives.” arXiv:1508.03267 [math], August 13, 2015. http://arxiv.org/abs/1508.03267.
* [http://navercast.naver.com/science/list 네이버 오늘의과학]
+
* Kowalenko, Victor. “Exactification of Stirling’s Approximation for the Logarithm of the Gamma Function.” arXiv:1404.2705 [math], April 10, 2014. http://arxiv.org/abs/1404.2705.
* [http://math.dongascience.com/ 수학동아]
+
* Connon, Donal F. “Fourier Series Representations of the Logarithms of the Euler Gamma Function and the Barnes Multiple Gamma Functions.” arXiv:0903.4323 [math], March 25, 2009. http://arxiv.org/abs/0903.4323.
* [http://www.ams.org/mathmoments/ Mathematical Moments from the AMS]
+
* Amdeberhan, Tewodros, Mark W. Coffey, Olivier Espinosa, Christoph Koutschan, Dante V. Manna, and Victor H. Moll. “Integrals of Powers of Loggamma.” Proceedings of the American Mathematical Society 139, no. 2 (2011): 535–45. doi:10.1090/S0002-9939-2010-10589-0. ,http://www.math.tulane.edu/~vhm/papers_html/log-gamma.pdf
* [http://betterexplained.com/ BetterExplained]
+
* Koyama, Shin-ya, and Nobushige Kurokawa. "Kummer's formula for multiple gamma functions." JOURNAL-RAMANUJAN MATHEMATICAL SOCIETY 18.1 (2003): 87-107. http://www.math.titech.ac.jp/~tosho/Preprints/pdf/128.pdf
 +
* Berndt, Bruce C. “The Gamma Function and the Hurwitz Zeta-Function.” The American Mathematical Monthly 92, no. 2 (1985): 126–30. doi:10.2307/2322640.

2020년 12월 28일 (월) 03:17 기준 최신판

개요

후르비츠 제타함수



적분표현

  • Binet's second expression\[\operatorname{Re} z > 0 \] 일 때, \(\log \Gamma(z)=(z-\frac{1}{2})\log z -z+\frac{1}{2}\log 2\pi+ 2\int_0^{\infty}\frac{\tan^{-1}(t/z)}{e^{2\pi t} -1}dt\)http://dlmf.nist.gov/5/9/ 참고



쿰머의 푸리에 급수

  • 쿰머 (1847)\[\begin{eqnarray}\log\Gamma(x)=\log\sqrt{2\pi}-\frac{1}{2}\log(2\sin\pi x)+\frac{1}{2}(\gamma+2\log\sqrt{2\pi})(1-2x)+\frac{1}{\pi}\sum_{k=1}^{\infty}\frac{\log k}{k}\sin 2\pi kx \nonumber \\ =(\frac{1}{2}-x)(\gamma+\log 2)+(1-x)\log \pi -\frac{1}{2}\log(\sin\pi x)+\frac{1}{\pi}\sum_{k=1}^{\infty}\frac{\log k}{k}\sin 2\pi kx \nonumber \end{eqnarray} \]



테일러 급수



정적분

\(\int_{0}^{1}\log\Gamma(x)\,dx=\log\sqrt{2\pi}\)


\(\int_{0}^{\frac{1}{2}}\log\Gamma(x+1)\,dx=-\frac{1}{2}-\frac{7}{24}\log 2+\frac{1}{4}\log \pi+\frac{3}{2}\log A\)

A는 Glaisher–Kinkelin 상수



스털링 공식





메모

관련된 항목들




사전 형태의 자료


관련논문

  • Diamond, Harold G., and Armin Straub. “Bounds for the Logarithm of the Euler Gamma Function and Its Derivatives.” arXiv:1508.03267 [math], August 13, 2015. http://arxiv.org/abs/1508.03267.
  • Kowalenko, Victor. “Exactification of Stirling’s Approximation for the Logarithm of the Gamma Function.” arXiv:1404.2705 [math], April 10, 2014. http://arxiv.org/abs/1404.2705.
  • Connon, Donal F. “Fourier Series Representations of the Logarithms of the Euler Gamma Function and the Barnes Multiple Gamma Functions.” arXiv:0903.4323 [math], March 25, 2009. http://arxiv.org/abs/0903.4323.
  • Amdeberhan, Tewodros, Mark W. Coffey, Olivier Espinosa, Christoph Koutschan, Dante V. Manna, and Victor H. Moll. “Integrals of Powers of Loggamma.” Proceedings of the American Mathematical Society 139, no. 2 (2011): 535–45. doi:10.1090/S0002-9939-2010-10589-0. ,http://www.math.tulane.edu/~vhm/papers_html/log-gamma.pdf
  • Koyama, Shin-ya, and Nobushige Kurokawa. "Kummer's formula for multiple gamma functions." JOURNAL-RAMANUJAN MATHEMATICAL SOCIETY 18.1 (2003): 87-107. http://www.math.titech.ac.jp/~tosho/Preprints/pdf/128.pdf
  • Berndt, Bruce C. “The Gamma Function and the Hurwitz Zeta-Function.” The American Mathematical Monthly 92, no. 2 (1985): 126–30. doi:10.2307/2322640.