\subsection{\bf Dangers in Integrating}
\subsection{\bf Non-Analytic Functions}

\begin{slide}

\begin{itemize}
\item (iv) Further Examples of Practical Technique

\item a) $I=\int^x_0 x^\alpha$ exists for $\alpha > -1$.
Namely, $I= {x^{\alpha + 1}\over{\alpha + 1}}$, 
but $f(x)=x^\alpha$ is not a polynomial unless $\alpha$ 
is an integer.  

\item In typical integration formula
$$
I = {h\over 3}\left\lbrack f(0) + 4f(h) + 2f(2h) + \ldots
\right\rbrack + 0(h^4)
$$

The error is $0\left(h^4 f^{(iv)}\right)$.  

\item Clearly this numerical procedure doesn't work if $\alpha < 0$
even though integral well defined as $-1 < \alpha < 0 \Rightarrow f(0)$
infinite, but $I$ well defined.
\vspace{10pt}

\item We could use an ``open'' formula which does not involved
$f(0)$ but even then, errors will be large.  For example $\int^h_0
f(x)\,dx$ has error something like $h^5\,f^{(iv)} (\zeta)$ where
$\zeta$ in range $0\le x\le h$.  If $f(x)=x^\alpha$, $f^{(iv)}
(\zeta)\sim\zeta^{\alpha -4}, \zeta\sim h$ implies that error is
$h^{1+\alpha}$ for each interval.  
\end{itemize}
\end{slide}

