"더블감마함수와 반스(Barnes) G-함수"의 두 판 사이의 차이
둘러보기로 가기
검색하러 가기
Pythagoras0 (토론 | 기여) |
Pythagoras0 (토론 | 기여) |
||
| 1번째 줄: | 1번째 줄: | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
==개요== | ==개요== | ||
| 15번째 줄: | 9번째 줄: | ||
| − | == | + | ==점근급수== |
| − | + | * [[점근 급수(asymptotic series)]] | |
| − | <math>\log G(z+1)=\frac{1}{12}~-~\log A~+~\frac{z}{2}\log 2\pi~+~\left(\frac{z^2}{2} -\frac{1}{12}\right)\log z~-~\frac{3z^2}{4}~+~ \sum_{k=1}^{N}\frac{B_{2k + 2}}{4k\left(k + 1\right)z^{2k}}~+~O\left(\frac{1}{z^{2N + 2}}\right)</math> | + | :<math>\log G(z+1)=\frac{1}{12}~-~\log A~+~\frac{z}{2}\log 2\pi~+~\left(\frac{z^2}{2} -\frac{1}{12}\right)\log z~-~\frac{3z^2}{4}~+~ \sum_{k=1}^{N}\frac{B_{2k + 2}}{4k\left(k + 1\right)z^{2k}}~+~O\left(\frac{1}{z^{2N + 2}}\right)</math> |
여기서 A는 [[Glaisher–Kinkelin 상수]] <math>A= e^{\frac{1}{12}-\zeta^\prime(-1)}= 1.28242712\dots</math> | 여기서 A는 [[Glaisher–Kinkelin 상수]] <math>A= e^{\frac{1}{12}-\zeta^\prime(-1)}= 1.28242712\dots</math> | ||
| 37번째 줄: | 31번째 줄: | ||
==로그 삼각함수 적분과의 관계== | ==로그 삼각함수 적분과의 관계== | ||
| − | <math>\int_{0}^{t}\pi u \cot \pi u\,du=t\log {2\pi}+\log \frac{G(1-t)}{G(1+t)}</math> | + | :<math>\int_{0}^{t}\pi u \cot \pi u\,du=t\log {2\pi}+\log \frac{G(1-t)}{G(1+t)}</math> |
| − | + | :<math>\int_{0}^{t}\log(\sin \pi u)\,du=t\log(\frac{\sin \pi t}{2\pi})+\log \frac{G(1+t)}{G(1-t)}</math> | |
| − | <math>\int_{0}^{t}\log(\sin \pi u)\,du=t\log(\frac{\sin \pi t}{2\pi})+\log \frac{G(1+t)}{G(1-t)}</math> | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| 59번째 줄: | 42번째 줄: | ||
==역사== | ==역사== | ||
| − | |||
| − | |||
| − | |||
| − | |||
* [[수학사 연표]] | * [[수학사 연표]] | ||
| − | |||
| − | |||
| − | |||
| 87번째 줄: | 63번째 줄: | ||
==수학용어번역== | ==수학용어번역== | ||
| − | + | * {{학술용어집|url=hyperfactorial}} | |
| − | * | ||
* 발음사전 http://www.forvo.com/search/Barnes | * 발음사전 http://www.forvo.com/search/Barnes | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
==사전 형태의 자료== | ==사전 형태의 자료== | ||
| − | |||
| − | |||
* http://en.wikipedia.org/wiki/Barnes_G-function | * http://en.wikipedia.org/wiki/Barnes_G-function | ||
* http://www.wolframalpha.com/input/?i=Barnes+G-function | * http://www.wolframalpha.com/input/?i=Barnes+G-function | ||
* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]<br> | * [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]<br> | ||
** [http://dlmf.nist.gov/5.17 § 5.17. Barnes’ -Function (Double Gamma Function)] | ** [http://dlmf.nist.gov/5.17 § 5.17. Barnes’ -Function (Double Gamma Function)] | ||
| − | |||
| − | |||
| − | |||
| − | |||
| 120번째 줄: | 82번째 줄: | ||
* [http://projecteuclid.org/euclid.tjm/1270472992 A Proof of the Classical Kronecker Limit Formula]<br> | * [http://projecteuclid.org/euclid.tjm/1270472992 A Proof of the Classical Kronecker Limit Formula]<br> | ||
** Takuro SHINTANI. Source: Tokyo J. of Math. Volume 03, Number 2 (1980), 191-199<br> | ** Takuro SHINTANI. Source: Tokyo J. of Math. Volume 03, Number 2 (1980), 191-199<br> | ||
| − | + | * Barnes, E. W. 2013. “The Genesis of the Double Gamma Functions.” Proceedings of the London Mathematical Society S1-31 (1): 358. doi:10.1112/plms/s1-31.1.358. | |
| − | * | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
2013년 3월 6일 (수) 11:53 판
개요
- 더블 감마함수의 역수로 정의되는 함수
- 성질\[G(1)=1\]\[G(s+1) =\Gamma(s)G(s)\]
- 자연수 n에 대하여 다음이 성립한다\[G(n)=(n-1)!\times (n-2)! \times\cdots 2!\times 1!\]
점근급수
\[\log G(z+1)=\frac{1}{12}~-~\log A~+~\frac{z}{2}\log 2\pi~+~\left(\frac{z^2}{2} -\frac{1}{12}\right)\log z~-~\frac{3z^2}{4}~+~ \sum_{k=1}^{N}\frac{B_{2k + 2}}{4k\left(k + 1\right)z^{2k}}~+~O\left(\frac{1}{z^{2N + 2}}\right)\]
여기서 A는 Glaisher–Kinkelin 상수 \(A= e^{\frac{1}{12}-\zeta^\prime(-1)}= 1.28242712\dots\)
special values
- A는 Glaisher–Kinkelin 상수\[G(\frac{1}{2})=2^{\frac{1}{24}}\cdot \pi^{-\frac{1}{4}}\cdot e^{\frac{1}{8}}\cdot A^{-\frac{3}{2}}\]\[G(\frac{3}{4})=2^{-\frac{1}{8}}\cdot \pi^{-\frac{1}{4}}\cdot e^{\frac{1}{8}}\cdot A^{-\frac{3}{2}}\] 또는 \(G(\frac{3}{4})=2^{-\frac{1}{8}}\cdot \pi^{-\frac{1}{4}}\cdot e^{\frac{3}{32}+\frac{G}{4\pi}}\cdot A^{-\frac{9}{8}}\cdot \Gamma(\frac{1}{4})^{\frac{1}{4}}\)
로그 삼각함수 적분과의 관계
\[\int_{0}^{t}\pi u \cot \pi u\,du=t\log {2\pi}+\log \frac{G(1-t)}{G(1+t)}\] \[\int_{0}^{t}\log(\sin \pi u)\,du=t\log(\frac{\sin \pi t}{2\pi})+\log \frac{G(1+t)}{G(1-t)}\]
역사
메모
관련된 항목들
수학용어번역
- hyperfactorial - 대한수학회 수학용어집
- 발음사전 http://www.forvo.com/search/Barnes
사전 형태의 자료
- http://en.wikipedia.org/wiki/Barnes_G-function
- http://www.wolframalpha.com/input/?i=Barnes+G-function
- NIST Digital Library of Mathematical Functions
관련논문
- Multiple Gamma and Related Functions
- J. Choi, H. M. Srivastava, V.S. Adamchik , Applied Mathematics and Computation, 134 (2003), 515-533
- A Proof of the Classical Kronecker Limit Formula
- Takuro SHINTANI. Source: Tokyo J. of Math. Volume 03, Number 2 (1980), 191-199
- Takuro SHINTANI. Source: Tokyo J. of Math. Volume 03, Number 2 (1980), 191-199
- Barnes, E. W. 2013. “The Genesis of the Double Gamma Functions.” Proceedings of the London Mathematical Society S1-31 (1): 358. doi:10.1112/plms/s1-31.1.358.