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Created page with 'The '''Gamma funciton''' is defined as <math>\Gamma(z) = \int_{0}^{\infty}t^{z-1}e^{-t}dt</math> Using integration by parts, <math>\Gamma(z) = \int_{…'
The '''Gamma funciton''' is defined as
<math>\Gamma(z) = \int_{0}^{\infty}t^{z-1}e^{-t}dt</math>
Using [[Integration_by_parts|integration by parts]],
<math>\Gamma(z) = \int_{0}^{\infty}t^{z-1}d(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) \Gamma(z-1)</math>
Thus,
<math> \Gamma (z) = (z-1) \Gamma(z-1)</math>
<math>\Gamma(z) = \int_{0}^{\infty}t^{z-1}e^{-t}dt</math>
Using [[Integration_by_parts|integration by parts]],
<math>\Gamma(z) = \int_{0}^{\infty}t^{z-1}d(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) \Gamma(z-1)</math>
Thus,
<math> \Gamma (z) = (z-1) \Gamma(z-1)</math>