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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 ,  \[
\begin{array}{rcl}
f^{(0)} & = & D ( x^{(0)} )
\\
e^{(j)} 
& = & d^{(j)} + \sum_{k=0}^{j} a^{(j-k)} * f^{(k)}
\\
& = & d^{(j)}
\\
f^{(j+1)} & = & \frac{1}{j+1} \frac{1}{ b^{(0)} } 
\left(
     \sum_{k=1}^{j+1} k x^{(k)} e^{(j+1-k)} 
     - \sum_{k=1}^j k f^{(k)}  b^{(j+1-k)} 
\right)
\\
& = & \frac{1}{j+1} 
     \sum_{k=1}^{j+1} k x^{(k)} d^{(j+1-k)} 
\end{array}
\] 
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) ] .
Input File: omh/sin_cos_forward.omh