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

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.

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