"Gamma function"의 두 판 사이의 차이
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(새 문서: \( \Gamma (n)=(n-1)! \) where \(n\) is a positive integer. \(\displaystyle \Gamma (z)=\int _{0}^{\infty }x^{z-1}e^{-x}\,dx\) for complex numbers with a positive real part. \(\disp...) |
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| + | 역시 위키[https://en.wikipedia.org/wiki/Gamma_function]는 훌륭하다. | ||
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\( \Gamma (n)=(n-1)! \) where \(n\) is a positive integer. | \( \Gamma (n)=(n-1)! \) where \(n\) is a positive integer. | ||
2017년 6월 19일 (월) 15:45 기준 최신판
역시 위키[1]는 훌륭하다.
\( \Gamma (n)=(n-1)! \) where \(n\) is a positive integer.
\(\displaystyle \Gamma (z)=\int _{0}^{\infty }x^{z-1}e^{-x}\,dx\) for complex numbers with a positive real part.
\(\displaystyle \Gamma (z+1)=\int _{0}^{\infty }x^{z}e^{-x}\,dx\)
\(\displaystyle =\left[-x^{z}e^{-x}\right]_{0}^{\infty }+\int _{0}^{\infty }zx^{z-1}e^{-x}\,dx\)
\(\displaystyle =\lim _{x\to \infty }(-x^{z}e^{-x})-(0e^{-0})+z\int _{0}^{\infty }x^{z-1}e^{-x}\,dx\)
Recognizing that as \(\displaystyle x\to \infty ,-x^{z}e^{-x}\to 0,\)
\(\displaystyle =z\int _{0}^{\infty }x^{z-1}e^{-x}\,dx=z\Gamma (z)\)