\(\newcommand{\W}[1]{ \; #1 \; }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }\) \(\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }\) \(\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }\) \(\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }\)
erf_forward
Error Function Forward Taylor Polynomial Theory
Derivatives
Given \(X(t)\), we define the function
It follows that
where we define the function
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
Setting the coefficients of \(t^{j-1}\) equal, we have