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:<math>= \left[t^{z-1}(-e^{-t}) \right]_0^{\infty}- \int_{0}^{\infty}(z-1)t^{(z-2)}(-e^{-t})dt</math>
:<math>= \left[t^{z-1}(-e^{-t}) \right]_0^{\infty}- \int_{0}^{\infty}(z-1)t^{(z-2)}(-e^{-t})dt</math>
:<math>=(z-1) \int_{0}^{\infty}t^{(z-1)-1}e^{-t}dt</math>
:<math>=(z-1) \int_{0}^{\infty}t^{(z-1)-1}e^{-t}dt</math>
:<math>=(z-1) \Gamma(z-1)</math>
:<math>=(z-1) \Gamma(z-1).</math>
Thus,
Thus,
:<math> \Gamma (z) = (z-1) \Gamma(z-1)</math>
:<math> \Gamma (z) = (z-1) \Gamma(z-1)</math>
:<math> \Gamma(n) = (n-1)!. </math>
:<math> \Gamma(n) = (n-1)!. </math>
[[category:mathematics]]