\(\newcommand{\W}[1]{ \; #1 \; }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }\) \(\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }\) \(\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }\) \(\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }\)
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: