Results and Formulas of Beta and Gamma Integrals

1)

\beta (m.n) = \int\limits_0^1 {{x^{m - 1}}{{(1 -  x)}^{n - 1}}dx}

It is called Beta Integral.

2)

\Gamma (x) = \int\limits_0^\infty {{e^{ - t}}{\text{ }}{t^{x - 1}}dt}

It is called 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 (n - 1) = \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 approximation value of

\Gamma (n + 1) = \sqrt  {2\pi n} {\text{ }}{n^n}{e^{ - n}}

It is called 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}}}}}

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