$Id: exp_forward.omh 3683 2015-05-10 02:24:16Z bradbell $
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CppAD: C++ Algorithmic Differentiation: Copyright (C) 2003-15 Bradley M. Bell

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$begin exp_forward$$
$spell
	exp
	expm1
	Taylor
$$

$section Exponential Function Forward Mode Theory$$
$mindex exp, expm1$$

$head Derivatives$$
If $latex F(x)$$ is $latex \R{exp} (x)$$ or $latex \R{expm1} (x)$$
the corresponding derivative satisfies the equation
$latex \[
1 * F^{(1)} (x) - 1 * F (x)
=
d^{(0)}
=
\left\{ \begin{array}{ll}
	0 & \R{if} \; F(x) = \R{exp}(x)
	\\
	1 & \R{if} \; F(x) = \R{expm1}(x)
\end{array} \right.
\] $$
where the equation above defines $latex d^{(0)}$$.
In the
$cref/standard math function differential equation
	/ForwardTheory
	/Standard Math Functions
	/Differential Equation
/$$,
$latex A(x) = 1$$,
$latex B(x) = 1$$,
and $latex D(x) = d^{(0)}$$.
We use $latex a$$, $latex b$$, $latex d$$,
and $latex z$$ to denote the
Taylor coefficients for
$latex A [ X (t) ] $$,
$latex B [ X (t) ]$$,
$latex D [ X (t) ] $$,
and $latex F [ X(t) ] $$ respectively.

$head Taylor Coefficients Recursion$$
For orders $latex j = 0 , 1, \ldots$$,
$latex \[
\begin{array}{rcl}
z^{(0)} & = & F ( x^{(0)} )
\\
e^{(0)} & = & d^{(0)} + z^{(0)}
\\
e^{(j+1)}
& = & d^{(j+1)} + \sum_{k=0}^{j+1} a^{(j+1-k)} * z^{(k)}
\\
& = & z^{(j+1)}
\\
z^{(j+1)} & = & \frac{1}{j+1} \frac{1}{ b^{(0)} }
\left(
	\sum_{k=1}^{j+1} k x^{(k)} e^{(j+1-k)}
	- \sum_{k=1}^j k z^{(k)}  b^{(j+1-k)}
\right)
\\
& = & x^{(j+1)} d^{(0)}
	+ \frac{1}{j+1} \sum_{k=1}^{j+1} k x^{(k)} z^{(j+1-k)}
\end{array}
\] $$

$end