Difference between revisions of "Gamma(1)=1"
From specialfunctionswiki
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==References== | ==References== | ||
| − | * {{BookReference|Special Functions for Scientists and Engineers|1968|W.W. Bell|prev=Beta|next=Gamma(z+1)=zGamma(z)}}: | + | * {{BookReference|Special Functions for Scientists and Engineers|1968|W.W. Bell|prev=Beta|next=Gamma(z+1)=zGamma(z)}}: Theorem 2.1 |
[[Category:Theorem]] | [[Category:Theorem]] | ||
[[Category:Proven]] | [[Category:Proven]] | ||
Latest revision as of 19:47, 15 March 2018
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): Theorem 2.1
