The definition of the gamma function is
\begin{align*}
\Gamma(p) = \int_{0}^{\infty} x^{p-1} e^{-x} dx
\end{align*}
where p > 0.
Characteristics of the gamma function.
\begin{align*}
\Gamma(p+1) &= p\Gamma(p) \\
\Gamma(1) &= 1 \\
\Gamma(\frac{1}{2}) &= \sqrt{\pi} \\
\Gamma(\frac{n}{2}) &= \begin{cases}
\left( \frac{n}{2} -1 \right)! ,& \text{if } n \text{ is even}\\
\left( \frac{n}{2} -1 \right) \left( \frac{n}{2} -2 \right) \cdots \frac{3}{2} \cdot \frac{1}{2} \sqrt{\pi} ,& \text{else n is odd and } n \ge 3
\end{cases}
\end{align*}
Let's prove these four characteristics.
\begin{align*}
\Gamma(p+1) &= \int_{0}^{\infty} x^{(p+1)-1} e^{-x} dx \\
&= \int_{0}^{\infty} x^{p} e^{-x} dx \\
&= \int_{0}^{\infty} x^{p} (- e^{-x})' dx \\
&= \left[ - x^{p} e^{-x} \right]_{0}^{\infty} - \int_{0}^{\infty} (x^{p})' (- e^{-x}) dx \\
&= p \int_{0}^{\infty} x^{p-1} e^{-x} dx \\
&= p \Gamma(p) \\
\end{align*}
\begin{align*}
\Gamma(1) &= \int_{0}^{\infty} x^{0} e^{-x} dx \\
&= \int_{0}^{\infty} e^{-x} dx \\
&= \left[ -e^{-x} \right]_{0}^{\infty} \\
&= 1
\end{align*}
\begin{align*}
\Gamma(\frac{1}{2}) &= \int_{0}^{\infty} x^{- \frac{1}{2}} e^{-x} dx \\
\end{align*}
Let t = x^{\frac{1}{2}} \iff x = t^2, dx/dt = 2t.
\begin{align*}
&= \int_{0}^{\infty} t^{-1} e^{-t^2} 2t dt \\
&= 2 \int_{0}^{\infty} e^{-t^2} dt \\
&= 2 \frac{\sqrt{\pi}}{2} \\
&= \sqrt{\pi}
\end{align*}
When n is even, let n = 2k , k \in \mathbb{N}.
\begin{align*}
\Gamma(\frac{n}{2}) &= \Gamma(\frac{2k}{2}) \\
&= \Gamma(k) \\
&= (k-1)! \\
&= \left( \frac{n}{2} -1 \right) ! \\
\end{align*}
When n is odd, let n = 2k - 1 , k \in \mathbb{N}.
\begin{align*}
\Gamma(\frac{n}{2}) &= \Gamma(\frac{2k+1}{2}) \\
&= \Gamma(k + \frac{1}{2}) \\
&= (k-1)! \Gamma(\frac{1}{2}) \\
&= \left( \frac{n}{2} -1 \right) ! \sqrt{\pi}\\
\end{align*}
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