Tag: Functional Equations

Use integration by parts to prove the functional equation of the Gamma function

by RoRi

Recall the definition of the Gamma function:

$\Gamma(s) = \int_{0^+}^{\infty} t^{s-1} e^{-t} \, dt \qquad \text{for} \quad s > 0.$

Using integration by parts, prove the functional equation:

$\Gamma(s+1) = s \Gamma(s).$

Then use mathematical induction to prove that for positive integers $n$ we have

$\Gamma(n+1) = n!.$

Incomplete.