# Results and Formulas of Beta and Gamma Integrals

1) $\beta (m.n) = \int\limits_0^1 {{x^{m – 1}}{{(1 – x)}^{n – 1}}dx}$ is called the Beta Integral.

2) $\Gamma (x) = \int\limits_0^\infty {{e^{ – t}}{\text{ }}{t^{x – 1}}dt}$ is called the Gamma Integral.

3) $\beta (m,n) = \beta (n,m)$

4) $\beta (m,n) = \frac{{\Gamma (m)\Gamma (n)}}{{\Gamma (m + n)}}$

5) $\Gamma (1) = \Gamma (2) = 1$

6) $\Gamma \left( {\frac{1}{2}} \right) = \sqrt \pi$

7) $\Gamma (n) = (n – 1){\text{!}}$ Where $n$ is a positive integer.

8) $\Gamma (n)\Gamma (1 – n) = \frac{\pi }{{\sin n\pi }}$ If $n$ is not an integer on $0 < n < 1$

9) $\Gamma \left( {n + \frac{1}{2}} \right) = 1 \cdot 3 \cdot 5 \cdot \cdots \cdot (2n – 1)\frac{{\sqrt \pi }}{{2n}}$

10) When $n$ is large and positive, then the approximation value of $\Gamma (n + 1) = \sqrt {2\pi n} {\text{ }}{n^n}{e^{ – n}}$ is called the Stirrling Formula.

11) $\int\limits_0^{\frac{\pi }{2}} {{{\sin }^n}xdx = } \int\limits_0^{\frac{\pi }{2}} {{{\cos }^n}xdx = \frac{{\sqrt \pi }}{2}} \frac{{\Gamma \left( {\frac{{n + 1}}{2}} \right)}}{{\Gamma \left( {\frac{n}{2} + 1} \right)}},{\text{ for }}n > – 1$

12) $\int\limits_0^\infty {{x^n}{e^{ – mx}}dx = \frac{{\Gamma (n + 1)}}{{{m^{n + 1}}}},{\text{ }}m > 0,{\text{ }}n > – 1}$

13) $\int\limits_0^\infty {{x^n}{e^{ – {m^2}{x^2}}}dx = \frac{{\Gamma \left( {\frac{{n + 1}}{2}} \right)}}{{2{{(m)}^{n + 1}}}}}$