SF3

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Show that \Gamma(n+1) = n \Gamma(n), n>0\,

\Gamma(n+1)= \int_0^\infty x^n e^{-x} dx = \lim_{M \to \infty} \int_0^M x^n e^{-x} dx \,

= \lim_{M \to \infty} \left[ \left[x^n(-e^{-x})\right]_{x=0}^M - \int_0^M(-e^{-x}(nx^{n-1})dx \right]

= \lim_{M\to \infty} \left[-M^ne^{-M}+n\int_0^Mx^{n-1}e^{-x}dx\right]\,

= n\int_0^\infty x^{n-1}e^{-x}dx\,

= n\Gamma(n)\,

Special Functions

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