"더블감마함수와 반스(Barnes) G-함수"의 두 판 사이의 차이
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==개요== | ==개요== | ||
| − | * 더블 감마함수의 역수로 정의되는 함수 | + | * 더블 감마함수의 역수로 정의되는 함수 |
| − | * 성질:<math>G(1)=1</math>:<math>G(s+1) =\Gamma(s)G(s)</math | + | * 성질:<math>G(1)=1</math>:<math>G(s+1) =\Gamma(s)G(s)</math> |
| − | * 자연수 n에 대하여 다음이 성립한다:<math>G(n)=(n-1)!\times (n-2)! \times\cdots 2!\times 1!</math | + | * 자연수 n에 대하여 다음이 성립한다:<math>G(n)=(n-1)!\times (n-2)! \times\cdots 2!\times 1!</math> |
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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> | ||
| − | * [[스털링 공식]] | + | * [[스털링 공식]] |
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==special values== | ==special values== | ||
| − | * A는 [[Glaisher–Kinkelin 상수]]:<math>G(\frac{1}{2})=2^{\frac{1}{24}}\cdot \pi^{-\frac{1}{4}}\cdot e^{\frac{1}{8}}\cdot A^{-\frac{3}{2}}</math>:<math>G(\frac{3}{4})=2^{-\frac{1}{8}}\cdot \pi^{-\frac{1}{4}}\cdot e^{\frac{1}{8}}\cdot A^{-\frac{3}{2}}</math> 또는 <math>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}}</math | + | * A는 [[Glaisher–Kinkelin 상수]]:<math>G(\frac{1}{2})=2^{\frac{1}{24}}\cdot \pi^{-\frac{1}{4}}\cdot e^{\frac{1}{8}}\cdot A^{-\frac{3}{2}}</math>:<math>G(\frac{3}{4})=2^{-\frac{1}{8}}\cdot \pi^{-\frac{1}{4}}\cdot e^{\frac{1}{8}}\cdot A^{-\frac{3}{2}}</math> 또는 <math>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}}</math> |
| 54번째 줄: | 54번째 줄: | ||
==관련된 항목들== | ==관련된 항목들== | ||
| − | * [[감마함수]] | + | * [[감마함수]] |
| − | * [[멀티 감마함수(multiple gamma function)]] | + | * [[멀티 감마함수(multiple gamma function)]] |
| − | * [[로그 사인 적분 (log sine integrals)]] | + | * [[로그 사인 적분 (log sine integrals)]] |
| 71번째 줄: | 71번째 줄: | ||
* 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] | + | * [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions] |
** [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)] | ||
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==관련논문== | ==관련논문== | ||
| − | * [http://www.cs.cmu.edu/~adamchik/articles/Srivastava/ch_sr.pdf Multiple Gamma and Related Functions] | + | * [http://www.cs.cmu.edu/~adamchik/articles/Srivastava/ch_sr.pdf Multiple Gamma and Related Functions] |
** J. Choi, H. M. Srivastava, V.S. Adamchik , Applied Mathematics and Computation, 134 (2003), 515-533 | ** J. Choi, H. M. Srivastava, V.S. Adamchik , Applied Mathematics and Computation, 134 (2003), 515-533 | ||
| − | * [http://projecteuclid.org/euclid.tjm/1270472992 A Proof of the Classical Kronecker Limit Formula] | + | * [http://projecteuclid.org/euclid.tjm/1270472992 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. | * 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. | ||
[[분류:특수함수]] | [[분류:특수함수]] | ||
2020년 11월 12일 (목) 06:52 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
- 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.