Difference between revisions of "Gamma(z+1)=zGamma(z)"
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==References== | ==References== | ||
| − | * {{BookReference|Special Functions for Scientists and Engineers|1968|W.W. Bell|prev=Gamma(1)=1|next=Gamma(n+1)=n!}}: | + | * {{BookReference|Special Functions for Scientists and Engineers|1968|W.W. Bell|prev=Gamma(1)=1|next=Gamma(n+1)=n!}}: Theorem 2.2 |
[[Category:Theorem]] | [[Category:Theorem]] | ||
[[Category:Proven]] | [[Category:Proven]] | ||
[[Category:Justify]] | [[Category:Justify]] | ||
Revision as of 19:47, 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): Theorem 2.2
