I am familiar with the gamma function, but When I needed to simplify the bessel function of the 1st kind \(\displaystyle J_{\frac{1}{2}}\) I stuck on \(\displaystyle \Gamma (n+\frac{3}{2})=(n+\frac{1}{2}!)\).

Isn't \(\displaystyle n! = n \cdot (n-1) \cdot (n-2)\cdot \dots \cdot 2 \cdot 1\) where \(\displaystyle n\) is an integer. But what if \(\displaystyle n\) is not, namely \(\displaystyle n=1/2\).

I started with \(\displaystyle (n+\frac{1}{2}!)=(n+\frac{1}{2})(n+\frac{-1}{2})\dots(\frac{1}{2}!)\) but wasn't sure if I must stop at 1 or at \(\displaystyle (\frac{1}{2}!)\). But then I multiplied each term by 2 so I got \(\displaystyle 2^n(2n+1)(2n-1)\dots(\frac{1}{2}!)\)

WHAT NEXT??