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\begin{center}\textbf{Problem Set 10 \hspace{.7in} Real Analysis 1 \hspace{.7in} Due 12/9/16} \end{center}

\noindent Four of these problems will be graded, with each problem worth 5 points.  Clear and complete justification is required for full credit.  You are welcome to discuss these problems with anyone and everyone, but must write up your own final submission without reference to any sources other than the textbook and instructor. 

\begin{enumerate}

\item Show that if $x>0$ then $\Gamma (x+1) = x \Gamma(x)$.
\item If $n \in \mathbb{N}$, then $\Gamma(n) = (n-1)!$.
\item State (legibly!) the definition of the error function.
\item State (legibly!) the definition of the Bessel function of the first kind of order $p$.
\item What's cool about the set of equivalence classes under the relation on $\mathbb{R}$ given by $a \sim b \Leftrightarrow a-b \in \mathbb{Q}$?
\item Let \[f(x)=\begin{cases} 0 &\mathrm{if } \, x \in \mathbb{R} - \mathbb{Q}\\
1&\mathrm{if } \, x \in \mathbb{Q} \end{cases}\]
What is the Lebesgue integral of $f$ on $[0,1]$?
\item Give a good example of a Dedekind cut.
\item What problem(s) would you like to see on the final exam?

\end{enumerate}

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