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\title{The Gamma Function}
\date{March 29, 2021}

\begin{document}
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\dfn{
The gamma function is defined by $\displaystyle \boxed{\Gamma(\alpha) = \int_0^{\infty} x^{\alpha-1}e^{-x}\ dx}$
}

\prob{[Completion] Investigate the gamma function:
\begin{enumerate}[a)]
\item Find $\Gamma(1)$
\item Use IBP ($\int u dv = uv - \int v du$) to find $\Gamma(2)$
\item Show that for positive $\alpha$, $\Gamma(\alpha+1) = \alpha \Gamma(\alpha)$ (use IBP again)
\end{enumerate}
}

\cmt{
One consequence of the last fact is that for any natural number $n$, $\Gamma(n+1) = n!$ (where $n!$ is the factorial: $n! = n(n-1)(n-2)\dots(3)(2)1$).
Hence $\Gamma(x)$ can be thought of as a continuous version of the factorial; continuous because the function is defined for every positive real number (and some negative real numbers).
For example $\Gamma(1/2) = \sqrt{\pi}$.
}

\prob{[Completion]
Use the fact that $\Gamma(1/2) = \sqrt{\pi}$ to calculate the following:
\begin{enumerate}[a)]
\item $\Gamma(3/2)$
\item $\Gamma(-1/2)$
\end{enumerate}
}


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