អាំងតេក្រាលអឺលែ (ភាគ 1​ត)

អាំងតែក្រាលអឺលែកំរិត 2 (ប្រភេទទី​2)

អាំងតេក្រាល Euler ប្រភេទទីពីរជាអនុគមន៍ហ្គាមម៉ា  (Gamma) កំនត់ដោយៈ $\displaystyle \Gamma (a)=\int\limits_{0}^{+\infty}x^{a-1}e^{-x}dx; a>0$

លក្ខណៈនៃអនុគមន៍ Gamma

1. $\Gamma (a+1)=a\Gamma (a)$
2. $\Gamma (a+n)=(a+n-1)(a+n-2)...a\Gamma (a)$
3. $\displaystyle \Gamma (1)=\int\limits_{0}^{+\infty}e^{-x}dx=1$
4. $\displaystyle \Gamma (n+1)=n!$

ទំនាក់ទំនងរវាងអនុគមន៍ B និង $\Gamma$

$\displaystyle B(a, b)=\frac{\Gamma (a)\Gamma (b)}{\Gamma (a+b)}$

លក្ខណៈពិសេស

• $\displaystyle B(a, 1-a)=\frac{\Gamma (a)\Gamma (1-a)}{\Gamma (1)}\Rightarrow \Gamma (a)\Gamma (1-a)=\frac{\pi}{sina\pi}$;$(0
• $\displaystyle B(\frac{1}{2}, \frac{1}{2})=\frac{\Gamma (\frac{1}{2})\Gamma (\frac{1}{2})}{\Gamma (1)}\Rightarrow \Gamma (\frac{1}{2})\Gamma (\frac{1}{2})=\pi$
• $\displaystyle \int\limits_{0}^{+\infty}\frac{e^{-x}}{\sqrt{x}}dx=\Gamma (\frac{1}{2})=\sqrt{\pi}$
• $\displaystyle \int\limits_{0}^{+\infty}e^{-u^2}du=\frac{1}{2}\int\limits_{0}^{+\infty}\frac{e^{-u^2}d(u^2)}{\sqrt{u^2}}=\frac{\sqrt{\pi}}{2}$
• $\displaystyle \Gamma (n+\frac{1}{2})=(n-\frac{1}{2})(n-\frac{3}{2})...\frac{5}{2}.\frac{3}{2}.\frac{1}{2}\Gamma (\frac{1}{2})=\frac{1.3.5...(2n-1)}{2^n}.\sqrt{\pi}$
• $\displaystyle 2^{2x-1}\Gamma (x)\Gamma (x+\frac{1}{2})=\sqrt{\pi}\Gamma (2x); x>0$
• $\displaystyle \Gamma (x+\frac{1}{n})\Gamma (x+\frac{2}{n})...\Gamma (x+\frac{n-1}{n})=\sqrt{n^{1-2nx}(2\pi)^{n-1}}.\Gamma (nx)$