Euler and the factorial function

Question

I recently purchased H. M. Edwards' book entitled The Riemann Zeta Function. In the early pages of the volume, concerning the factorial function $\Gamma$, Edwards notes that

"Euler observed that $\Gamma(n)=\int_0^\infty e^{-x}x^{n-1}dx$."

My question is twofold:

1. How does one "observe" such a formula? Surely, this does not merely come from an intuitive observation?
2. How does one prove this formula, and more importantly, where does the techincal motivation for it come from?

Answer 1

If you have not seen integration by parts before, it is strongly related to the product rule of differentiation. $$\frac{d}{dx}x^ne^{-x}=nx^{n-1}e^{-x}-x^ne^{-x}\\ \left.x^ne^{-x}\right|_0^{\infty}=\int_0^{\infty}nx^{n-1}e^{-x}dx-\int_0^{\infty}x^ne^{-x}dx\\ \int_0^{\infty}x^ne^{-x}dx=n\int_0^{\infty}x^{n-1}e^{-x}dx$$ So the integral with $n$ is related to the integral with $n-1$; and by the same rule that $n!$ is.

Answer 2

(1) According to the Wikipedia, the definition of $\Gamma$ by Euler was $$\Gamma(x)=\lim_{n\to\infty}\frac{n! n^x}{x(x+1)\cdots(x+n)}.$$ (2) See Baby Rudin, 8.17,8.18,8.19 or Equivalence of Definitions of Gamma Function.