Difference between revisions of "Gamma(z+1)=zGamma(z)"
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m (Tom moved page Factorial property of gamma to Gamma(z+1)=zGamma(z): Gamma(z+1)=zGamma(z)) |
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| − | + | ==Theorem== | |
| − | + | The following formula holds: | |
| − | $$\Gamma( | + | $$\Gamma(z+1)=z\Gamma(z),$$ |
| − | where $\Gamma$ denotes | + | where $\Gamma$ denotes [[gamma]]. |
| − | + | ||
| − | + | ==Proof== | |
| + | Use [[integration by parts]] to compute | ||
$$\begin{array}{ll} | $$\begin{array}{ll} | ||
| − | \Gamma( | + | \Gamma(z+1) &= \displaystyle\int_0^{\infty} \xi^z e^{-\xi} \mathrm{d}\xi \\ |
| − | &= -\xi^ | + | &= -\xi^z e^{-\xi}\Bigg|_0^{\infty}- \displaystyle\int_0^{\infty} z \xi^{z-1} e^{-\xi} \mathrm{d}\xi \\ |
| − | &= | + | &= z\Gamma(z), |
\end{array}$$ | \end{array}$$ | ||
as was to be shown. █ | as was to be shown. █ | ||
| − | + | ||
| − | + | ==References== | |
| + | * {{BookReference|Special Functions for Scientists and Engineers|1968|W.W. Bell|prev=Gamma(1)=1|next=Gamma(n+1)=n!}}: $(2.2)$ | ||
| + | |||
| + | [[Category:Theorem]] | ||
| + | [[Category:Proven]] | ||
| + | [[Category:Justify]] | ||
Revision as of 19:46, 15 March 2018
Theorem
The following formula holds: $$\Gamma(z+1)=z\Gamma(z),$$ where $\Gamma$ denotes gamma.
Proof
Use integration by parts to compute $$\begin{array}{ll} \Gamma(z+1) &= \displaystyle\int_0^{\infty} \xi^z e^{-\xi} \mathrm{d}\xi \\ &= -\xi^z e^{-\xi}\Bigg|_0^{\infty}- \displaystyle\int_0^{\infty} z \xi^{z-1} e^{-\xi} \mathrm{d}\xi \\ &= z\Gamma(z), \end{array}$$ as was to be shown. █
References
- 1968: W.W. Bell: Special Functions for Scientists and Engineers ... (previous) ... (next): $(2.2)$
