$Id:$
/* --------------------------------------------------------------------------
CppAD: C++ Algorithmic Differentiation: Copyright (C) 2003-14 Bradley M. Bell

CppAD is distributed under multiple licenses. This distribution is under
the terms of the 
                    Eclipse Public License Version 1.0.

A copy of this license is included in the COPYING file of this distribution.
Please visit http://www.coin-or.org/CppAD/ for information on other licenses.
-------------------------------------------------------------------------- */

$begin erf_forward$$
$spell
	erf
	Taylor
$$

$index erf, forward theory$$
$index theory, erf forward$$
$index forward, erf theory$$

$section Error Function Forward Taylor Polynomial Theory$$

$head Derivatives$$
Given $latex X(t)$$, we define the function 
$latex \[
	Z(t) = \R{erf}[ X(t) ]
\]$$
It follows that
$latex \[
\begin{array}{rcl}
\R{erf}^{(1)} ( u ) & = & ( 2 / \sqrt{\pi} ) \exp \left( - u^2 \right)
\\
Z^{(1)} (t) & = & \R{erf}^{(1)} [ X(t) ] X^{(1)} (t) = Y(t) X^{(1)} (t)
\end{array}
\] $$
where we define the function 
$latex \[
	Y(t) = \frac{2}{ \sqrt{\pi} } \exp \left[ - X(t)^2 \right]
\] $$

$head Taylor Coefficients Recursion$$
Suppose that we are given the Taylor coefficients 
up to order $latex j$$ for the function $latex X(t)$$ and $latex Y(t)$$.
We need a formula that computes the coefficient of order $latex j$$
for $latex Z(t)$$.
Using the equation above for $latex Z^{(1)} (t)$$, we have
$latex \[
\begin{array}{rcl}
\sum_{k=1}^j k z^{(k)} t^{k-1}
& = &
\left[ \sum_{k=0}^j y^{(k)} t^k        \right]
\left[ \sum_{k=1}^j k x^{(k)} t^{k-1}  \right]
+
o( t^{j-1} )
\end{array}
\] $$
Setting the coefficients of $latex t^{j-1}$$ equal, we have
$latex \[
\begin{array}{rcl}
j z^{(j)} 
=
\sum_{k=1}^j k x^{(k)} y^{(j-k)}
\\
z^{(j)} 
=
\frac{1}{j} \sum_{k=1}^j k x^{(k)} y^{(j-k)}
\end{array}
\] $$ 

$end