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