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Error Function Forward Taylor Polynomial Theory

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

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