Difference between revisions of "Gamma(z+1)=zGamma(z)"

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==Theorem==
<strong>[[Factorial property of gamma|Theorem]]:</strong>
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The following formula holds:
$$\Gamma(x+1)=x\Gamma(x), \quad x>0,$$
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$$\Gamma(z+1)=z\Gamma(z),$$
where $\Gamma$ denotes the [[gamma]] function.
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where $\Gamma$ denotes [[gamma]].
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<strong>Proof:</strong> Use [[integration by parts]] to compute
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==Proof==
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Use [[integration by parts]] to compute
 
$$\begin{array}{ll}
 
$$\begin{array}{ll}
\Gamma(x+1) &= \displaystyle\int_0^{\infty} \xi^x e^{-\xi} \mathrm{d}\xi \\
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\Gamma(z+1) &= \displaystyle\int_0^{\infty} \xi^z e^{-\xi} \mathrm{d}\xi \\
&= -\xi^x e^{-\xi}|_0^{\infty} \displaystyle\int_0^{\infty} x \xi^{x-1} e^{-\xi} \mathrm{d}\xi \\
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&= -\xi^z e^{-\xi}\Bigg|_0^{\infty}- \displaystyle\int_0^{\infty} z \xi^{z-1} e^{-\xi} \mathrm{d}\xi \\
&= x\Gamma(x),
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&= z\Gamma(z),
 
\end{array}$$
 
\end{array}$$
 
as was to be shown. █
 
as was to be shown. █
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==References==
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* {{BookReference|Special Functions for Scientists and Engineers|1968|W.W. Bell|prev=Gamma(1)=1|next=Gamma(n+1)=n!}}: Theorem 2.2
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* {{BookReference|Hypergeometric Orthogonal Polynomials and Their q-Analogues|2010|Roelof Koekoek|author2=Peter A. Lesky|author3=René F. Swarttouw|prev=Gamma|next=findme}}: $(1.2.2)$
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* {{BookReference|Special functions, a graduate text|2010|Richard Beals|author2=Roderick Wong|prev=Gamma|next=findme}}: $(2.1.2)$
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[[Category:Theorem]]
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[[Category:Proven]]
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[[Category:Justify]]

Latest revision as of 18:12, 16 June 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