Gamma(1)=1

Theorem

The following formula holds: $$\Gamma(1)=1,$$ where $\Gamma$ denotes the gamma function.

Proof

Compute using the fundamental theorem of calculus, $$\begin{array}{ll} \Gamma(1) &= \displaystyle\int_0^{\infty} \xi^{0} e^{-\xi} \mathrm{d}\xi \\ &= \displaystyle\int_0^{\infty} e^{-\xi} \mathrm{d}\xi \\ &= \left[ -e^{-\xi} \right.\Bigg|_{0}^{\infty} \\ &= 1, \end{array}$$ as was to be shown. █

References

• 1968: W.W. Bell: Special Functions for Scientists and Engineers ... (previous) ... (next): $(2.1)$