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erf_forward

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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{eqnarray} \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{eqnarray}

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{eqnarray} \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{eqnarray}

Setting the coefficients of \(t^{j-1}\) equal, we have

\begin{eqnarray} 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{eqnarray}