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Exponential Function Forward Taylor Polynomial Theory
If  F(x) = \exp(x)   \[
     1 * F^{(1)} (x) - 1 * F (x)  = 0
\] 
and in the standard math function differential equation ,  A(x) = 1 ,  B(x) = 1 , and  D(x) = 0 . We use  a ,  b ,  d , and  z to denote the Taylor coefficients for  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 ,  \[
\begin{array}{rcl}
z^{(0)} & = & \exp ( x^{(0)} )
\\
e^{(j)} 
& = & d^{(j)} + \sum_{k=0}^{j} a^{(j-k)} * z^{(k)}
\\
& = & z^{(j)}
\\
z^{(j+1)} & = & \frac{1}{j+1} \frac{1}{ b^{(0)} } 
\left(
     \sum_{k=1}^{j+1} k x^{(k)} e^{(j+1-k)} 
     - \sum_{k=1}^j k z^{(k)}  b^{(j+1-k)} 
\right)
\\
& = & \frac{1}{j+1} 
     \sum_{k=1}^{j+1} k x^{(k)} z^{(j+1-k)} 
\end{array}
\] 

Input File: omh/exp_forward.omh