\(\newcommand{\W}[1]{ \; #1 \; }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }\) \(\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }\) \(\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }\) \(\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }\)
tan_forward
Tangent and Hyperbolic Tangent Forward Taylor Polynomial Theory
Derivatives
If \(F(u)\) is \(\tan (u)\) or \(\tanh (u)\) the corresponding derivative is given by
Given \(X(t)\), we define the function \(Z(t) = F[ X(t) ]\). It follows that
where we define the function \(Y(t) = Z(t)^2\).
Taylor Coefficients Recursion
Suppose that we are given the Taylor coefficients up to order \(j\) for the function \(X(t)\) and up to order \(j-1\) for the functions \(Y(t)\) and \(Z(t)\). We need a formula that computes the coefficient of order \(j\) for \(Y(t)\) and \(Z(t)\). Using the equation above for \(Z^{(1)} (t)\), we have
Setting the coefficients of \(t^{j-1}\) equal, we have
Once we have computed \(z^{(j)}\), we can compute \(y^{(j)}\) as follows: