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

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Trigonometric and Hyperbolic Sine and Cosine Forward Theory

Differential Equation

The standard math function differential equation is

\[B(u) * F^{(1)} (u) - A(u) * F (u) = D(u)\]

In this sections we consider forward mode for the following choices:

		\(F(u)\)		\(\sin(u)\)		\(\cos(u)\)		\(\sinh(u)\)		\(\cosh(u)\)
		\(A(u)\)		\(0\)		\(0\)		\(0\)		\(0\)
		\(B(u)\)		\(1\)		\(1\)		\(1\)		\(1\)
		\(D(u)\)		\(\cos(u)\)		\(- \sin(u)\)		\(\cosh(u)\)		\(\sinh(u)\)

We use \(a\), \(b\), \(d\) and \(f\) for the Taylor coefficients of \(A [ X (t) ]\), \(B [ X (t) ]\), \(D [ X (t) ]\), and \(F [ X(t) ]\) respectively. It now follows from the general Taylor Coefficients Recursion Formula that for \(j = 0 , 1, \ldots\),

\begin{eqnarray} f^{(0)} & = & D ( x^{(0)} ) \\ e^{(j)} & = & d^{(j)} + \sum_{k=0}^{j} a^{(j-k)} * f^{(k)} \\ & = & d^{(j)} \\ f^{(j+1)} & = & \frac{1}{j+1} \frac{1}{ b^{(0)} } \left( \sum_{k=1}^{j+1} k x^{(k)} e^{(j+1-k)} - \sum_{k=1}^j k f^{(k)} b^{(j+1-k)} \right) \\ & = & \frac{1}{j+1} \sum_{k=1}^{j+1} k x^{(k)} d^{(j+1-k)} \end{eqnarray}

The formula above generates the order \(j+1\) coefficient of \(F[ X(t) ]\) from the lower order coefficients for \(X(t)\) and \(D[ X(t) ]\).