Difference between revisions of "Gamma(1)=1"
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| Line 6: | Line 6: | ||
<strong>Proof:</strong> Compute | <strong>Proof:</strong> Compute | ||
$$\begin{array}{ll} | $$\begin{array}{ll} | ||
| − | \Gamma(1) &= \displaystyle\int_0^{\infty} \xi^{0} e^{-\xi} d\xi \\ | + | \Gamma(1) &= \displaystyle\int_0^{\infty} \xi^{0} e^{-\xi} \mathrm{d}\xi \\ |
| − | &= \displaystyle\int_0^{\infty} e^{-\xi} d\xi \\ | + | &= \displaystyle\int_0^{\infty} e^{-\xi} \mathrm{d}\xi \\ |
&= \left[ -e^{-\xi} \right]_{0}^{\infty} \\ | &= \left[ -e^{-\xi} \right]_{0}^{\infty} \\ | ||
&= 1, | &= 1, | ||
Revision as of 05:41, 16 May 2016
Theorem: The following formula holds: $$\Gamma(1)=1,$$ where $\Gamma$ denotes the gamma function.
Proof: Compute $$\begin{array}{ll} \Gamma(1) &= \displaystyle\int_0^{\infty} \xi^{0} e^{-\xi} \mathrm{d}\xi \\ &= \displaystyle\int_0^{\infty} e^{-\xi} \mathrm{d}\xi \\ &= \left[ -e^{-\xi} \right]_{0}^{\infty} \\ &= 1, \end{array}$$ as was to be shown. █
