Difference between revisions of "Gamma(1)=1"
From specialfunctionswiki
| (7 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
| − | + | ==Theorem== | |
| − | + | The following formula holds: | |
$$\Gamma(1)=1,$$ | $$\Gamma(1)=1,$$ | ||
where $\Gamma$ denotes the [[gamma]] function. | where $\Gamma$ denotes the [[gamma]] function. | ||
| − | + | ==Proof== | |
| − | + | Compute using the [[fundamental theorem of calculus]], | |
$$\begin{array}{ll} | $$\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 | + | &= \left[ -e^{-\xi} \right.\Bigg|_{0}^{\infty} \\ |
| − | &= 1 | + | &= 1, |
\end{array}$$ | \end{array}$$ | ||
as was to be shown. █ | as was to be shown. █ | ||
| − | + | ||
| − | + | ==References== | |
| + | * {{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: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
