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