Evaluating the Gamma function integral for an arbitrary value.

Question

How do you compute this integral?

$$\Gamma(\alpha ) =\int_{0}^{\infty}e^{-x}x^{\alpha -1}dx$$

I tried doing integration by parts but it continued to repeat.

Answer 1

Let $$I =\int_{0}^{\infty}e^{-x}x^{\alpha -1}dx$$ now, when you integrate by part taking $x^{a-1}$ to be second function and $e^{-x}$ first function you get,

$I= -x^{a-1}e^{-x} + \frac{1}{a}\int_{0}^{\infty}x^{a}e^{-x}dx$ now

$I= -x^{a-1}e^{-x} + \frac{1}{a}[(a-1)x^{a-1}e^{-x} + (a-1)\int_{0}^{\infty}x^{a-1}e^{-x}dx]$ so we can write this as

$I= -x^{a-1}e^{-x} + \frac{1}{a}[(a-1)x^{a-1}e^{-x} + (a-1)I]$ Now I think you can continue from here.