Definite Integrals
13.0 Gamma and Beta Functions
13.0 Gamma and Beta Functions
If $n$ is a positive rational number, then the Gamma Function is defined as a improper integral $$\int\limits_0^\infty {{e^{ - x}}{x^{n - 1}}dx} $$ It is denoted by $\Gamma \left( n \right)$.$$\Gamma \left( n \right) = \int\limits_0^\infty {{e^{ - x}}{x^{n - 1}}dx} $$
Properties of Gamma Function
(i) $\Gamma \left( {n + 1} \right) = n\Gamma \left( n \right) = n!$
(ii) $\Gamma \left( 1 \right) = 1$
(iii) $\Gamma \left( 0 \right) = \infty $
(iv) $\Gamma \left( {{1 \over 2}} \right) = \sqrt \pi $
Beta Function
The Beta Function is defined as $$\beta \left( {m,n} \right) = \int\limits_0^1 {{x^{m - 1}}{{\left( {1 - x} \right)}^{n - 1}}dx} $$ where $m,n >0$.
It can also be expressed in terms of trigonometric function as $$\beta \left( {m,n} \right) = 2\int\limits_0^{{\pi \over 2}} {{{\sin }^{2m - 1}}\theta {{\cos }^{2n - 1}}\theta d\theta } $$
Relation between Gamma and Beta Functions
$$\beta \left( {m,n} \right) = {{\Gamma \left( m \right) \cdot \Gamma \left( n \right)} \over {\Gamma \left( {m + n} \right)}}$$
Question 30. Find the value of $I = \int\limits_0^\infty {{e^{ - {x^3}}}\sqrt x dx} $
Solution: Let $x = {\left( t \right)^{{1 \over 3}}}$ $$dx = {1 \over 3}{t^{ - {2 \over 3}}}dt$$ Integral $I$ become $$I = \int\limits_0^\infty {{e^{ - t}} \cdot {t^{{1 \over 6}}} \cdot {1 \over 3} \cdot {t^{ - {2 \over 3}}}dt} $$ $$I = {1 \over 3}\int\limits_0^\infty {{e^{ - t}} \cdot {t^{{{ - 1} \over 2}}}dt} $$ $$I = {1 \over 3}\Gamma \left( {{1 \over 2}} \right)$$ $$I = {{\sqrt \pi } \over 3}$$
