How do I find the factorial of a decimal

Question

How do I find the following:

$$(0.5)!(-0.5)!$$

Can someone help me step by step here?

Answer 1

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: $$ \quad \Rightarrow \left(\dfrac{1}{2} \right)! \left( -\dfrac{1}{2} \right) ! = \dfrac {1}{2} \left( \Gamma ( \dfrac {1}{2} ) \right)^2 = \dfrac {\pi} {2} $$

Answer 2

Use the gamma function:

$$\Gamma(x)=\int_0^\infty e^{-t}t^{x-1}\,dt.$$

Answer 3

Gamma function the rescue! It is generalized of factorial to all non-negative even values $$(0.5)!=\Gamma(-0.5)=-2\sqrt{\pi}$$ $$(-0.5)!=\Gamma(-1.5)=\frac{4}{3}\sqrt{\pi}$$