Inverse Tangent and Hyperbolic Tangent Forward Mode Theory
Derivatives
@[@
\begin{array}{rcl}
\R{atan}^{(1)} (x) & = & 1 / ( 1 + x * x )
\\
\R{atanh}^{(1)} (x) & = & 1 / ( 1 - x * x )
\end{array}
@]@If @(@
F(x)
@)@ is @(@
\R{atan} (x)
@)@ or @(@
\R{atanh} (x)
@)@,
the corresponding derivative satisfies the equation
@[@
(1 \pm x * x ) * F^{(1)} (x) - 0 * F (x) = 1
@]@
and in the
standard math function differential equation
,
@(@
A(x) = 0
@)@,
@(@
B(x) = 1 \pm x * x
@)@,
and @(@
D(x) = 1
@)@.
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.