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\title{Symbolic Computation Minitoolbox}
\author{\jph}
\date{May 18, 2009}

\begin{document}                        \maketitle

This toolbox recognizes two \emph{types of variables}:
\begin{enumerate}
\item arguments to functions, which in this text are denotes by
  $x\eqdef\{x_1,x_2,\ldots, x_n\}$; and
\item constant parameters, which in this text are denotes by
  $x\eqdef\{c_1,c_2,\ldots, c_m\}$.
\end{enumerate}
This toolbox recognizes the following \emph{elementary functions} on the
scalar variable $x_k\in x$:
\begin{align*}
  p(x_k;c_i,c_j) &= (x_k-c_i)^{c_j}\\
  \delta(x_k;c_i) &=\begin{cases}
    1 & x_k=c_i\\
    0 & x_k\neq c_i,
  \end{cases}
\end{align*}
where $c_i,c_j\in c$ are scalar real parameters.

\medskip

The class of functions recognized by the toolbox consists of all
polynomials of elementary functions, i.e.,
\begin{align*}
  f(x)= \sum_i c_i \prod_j f_{ij}(x)^{c_{ij}},
\end{align*}
where $c_i,c_{ij}\in c$ and all the $f_{i j}$ are elementary functions.

\medskip

The following rules are used to take \emph{derivatives}
\begin{align*}
  \Deriv{p(x_\ell;c_i,c_j)}{x_k}
  &=\begin{cases}
    c_j(x_\ell-c_i)^{c_j-1} & \ell=k\\
    0 & \ell\neq k
  \end{cases}\\
  \Deriv{\delta(x_\ell;c_i)}{x_k}
  &=\begin{cases}
    \text{not defined} & \ell=k\\
    0 & \ell\neq k
  \end{cases}\\
  \Deriv{f(x)}{x_k}
  &= \sum_i c_i \sum_\ell c_{i\ell} f_{i\ell}(x)^{c_{i\ell}-1} \Deriv{f_{i\ell}(x)}{x_k} \prod_{j\neq\ell} f_{ij}(x)^{c_{ij}},
\end{align*}
The following rules are used to take \emph{finite differences}
\begin{align*}
  p(x_k+c_\ell;c_i,c_j)-p(x_k;c_i,c_j)
  &=(x_k+c_\ell-c_i)^{c_j}-(x_k-c_i)^{c_j}
  =\sum_{r=0}^{c_j-1} \binom{c_j}{r} c_\ell^{c_j-r} (x_k-c_i)^r\\
  \delta(x_k+c_\ell;c_i)-\delta(x_k;c_i)
  &=\delta(x_k;c_i-c_\ell)-\delta(x_k;c_i)\\
  f(x+\bar c)-f(x)
  &=\sum_i c_i \prod_j f_{ij}(x+\bar c)^{c_{ij}}-\sum_i c_i \prod_j f_{ij}(x)^{c_{ij}}
\end{align*}



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