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# Tag Archives: integration

## Gamma Function

If we consider the integral $I =\displaystyle{\int_0^{\infty}} e^{-t} t^{a-1} \mathrm dt$ , it is once seen to be an infinite and improper integral. This integral is infinite because the upper limit of integration is infinite and it is improper because $t=0$ is a point of infinite discontinuity of the integrand, if $a<1$, where $a$ is either real number or real part of a complex number. This integral is known as Euler’s Integral. This is of a great importance in mathematical analysis and calculus. The result, i.e., integral, is defined as a new function of real number $a$, as $\Gamma (a) =\displaystyle{\int_0^{\infty}} e^{-t} t^{a-1} \mathrm dt$ .

## Dirichlet’s Theorem and Liouville’s Extension of Dirichlet’s Theorem

### Topic

Beta & Gamma functions

### Statement

$\int \, \int \, \int_{V} \, x^{l-1} y^{m-1} z^{n-1} dx \, dy \,dz = \frac { \Gamma {(l)} \Gamma {(m)} \Gamma {(n)} }{ \Gamma{(l+m+n+1)} }$
where V is the region given by $x \ge 0, y \ge 0, z \ge 0, \, x+y+z \le 1$.