<s>[INST] How do I find the factorial of a decimal How do I find the following:
$$(0.5)!(-0.5)!$$
Can someone help me step by step here?
 [/INST] Factorial of any real number $n$ is defined by Gamma function as follows:
$$\Gamma (n) = (n-1)!$$
$$\quad \Rightarrow ( \dfrac{1}{2} )! ( -\dfrac{1}{2} ) ! = ( \dfrac{3}{2}-1 ) ! ( \dfrac{1}{2}-1 ) ! = \Gamma ( \dfrac {3} {2} ) \Gamma ( \dfrac {1}{2} )$$
It is also known that:
$$\Gamma {(1+z)} = z\Gamma {(z)}$$
$$\quad \Rightarrow \Gamma ( \dfrac {3} {2} ) \Gamma ( \dfrac {1}{2} ) = \dfrac {1}{2} \Gamma ( \dfrac {1}{2} )\Gamma ( \dfrac {1}{2} ) = \dfrac {1}{2} \left( \Gamma ( \dfrac {1}{2} ) \right)^2 $$
Since $\Gamma ( \dfrac {1}{2} ) = \sqrt{\pi}$, then we have:
$$ \quad \Rightarrow \left(\dfrac{1}{2} \right)! \left( -\dfrac{1}{2} \right) ! = \dfrac {1}{2} \left( \Gamma ( \dfrac {1}{2} ) \right)^2 = \dfrac {\pi} {2} $$
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