\(2).\) The improper integral \( \displaystyle \int_{0}^{\infty} e^{-x}x^{n-1}dx\) is convergent when \(n>0\), als0 this intergal is called gamma function i.e \(\displaystyle \int_{0}^{\infty} e^{-x}x^{n-1} = \Gamma(n)\)
\(3).\) \(B(1,1) = 1\)
\(4).\) \(B(m,n) = B(n,m)\)
\(5).\) \(B(m+1,n) = \displaystyle \frac{m}{m+1}B(m,n)\)
\(6).\) \(B(m,n) = \displaystyle 2\int_{0}^{\frac{\pi}{2}}\sin^{2m-1}\theta\cos^{2n-1} \theta \sin \theta \cos \theta d\theta,\) \(m>0,n>0\)
\(7).\) \(\displaystyle \frac{1}{2}B\left(\frac{m+1}{2},\frac{n+1}{2}\right) = \displaystyle \int_{0}^{\frac{\pi}{2}}\sin^{m}\theta\cos^{n} \theta \; d\theta\;\;\) where \(m>-1, n>-1\)
\(8).\) \(\displaystyle \frac{1}{2}B\left(\frac{n+1}{2},\frac{1}{2}\right) = \displaystyle \int_{0}^{\frac{\pi}{2}}\sin^{n} \theta = \displaystyle \int_{0}^{\frac{\pi}{2}}\cos^{n} \theta \; d\theta\;\;\) where \(n>-1\)
\(9).\) \(\displaystyle B\left(\frac{1}{2},\frac{1}{2}\right) = \displaystyle 2\int_{0}^{\frac{\pi}{2}}d\theta = \pi \;\;\)
\(10).\) \(\displaystyle B\left(m,n\right) = \displaystyle \int_{0}^{\infty}\frac{x^{m-1}}{(1+x)^{m+n}}dx, \;\; m>0,n>0\)
\(11).\) \(\displaystyle B\left(m,n\right) = \displaystyle \int_{0}^{1}\frac{x^{m-1} + x^{n-1}}{(1+x)^{m+n}}dx. \;\;\)
\(12).\) \(\Gamma(1) = 1\)
\(13).\) \(\Gamma(n+1) = n\Gamma(n),\;\; n>0\)
\(14).\) If \(n\) is a positive integer then \(\;\Gamma(n+1) = n!,\;\;\)
\(15).\) \(B(m,n) = \displaystyle \frac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}\) where \(m >0, n>0\).
\(16).\) \(\displaystyle \Gamma(\frac{1}{2}) = \sqrt{\pi}\).
\(17).\) \(B\left(\frac{1}{2},\frac{1}{2}\right)= \displaystyle (\Gamma(\frac{1}{2}))^2 = \pi\)
\(18).\) Legender's duplicate formula \(\sqrt{\pi} \Gamma(2n) = 2^{2n-1} \Gamma(n)\Gamma(n+\frac{1}{2})\), n>0
\(19).\) \(\displaystyle \Gamma(m)\Gamma(1-m) = \frac{\pi}{\sin m \pi}\) where \(0< m < 1 \).
\(20).\) \(\displaystyle \int_{0}^{\infty} e^{-kx}x^{n-1}dx = \frac{\Gamma(n)}{k^{n}}\). where k >0, n>0
\(21).\) \(\displaystyle \int_{1}^{\infty} \frac{(\log x)^{n-1}}{x^{k+1}}dx = \frac{\Gamma(n)}{k^{n}}\). where k >0, n>0
\(22).\) \(\displaystyle \int_{0}^{\infty} e^{-x^2}dx = \frac{1}{2}\sqrt{\pi}\).
\(23).\) \(\displaystyle \int_{-\infty}^{\infty} e^{-x^2}dx = \sqrt{\pi}\).
\(24).\) \(\displaystyle \zeta(s)\Gamma(s) = \int_{0}^{\infty} \frac{x^{s-1}}{e^x - 1}dx\), where \(\zeta (s) = \displaystyle \sum_{n=1}^{\infty} \frac{1}{n^s}\) is Riemann zeta function and s > 1
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