Positive Orders Z(t)
For order
j > 0
,
suppose that
H
is the same as
G
.
z^{(j)}
=
\frac{1}{j} \sum_{k=1}^j k x^{(k)} y^{(j-k)}
For
k = 1 , \ldots , j
,
the partial of
H
with respect to
x^{(k)}
is given by
\D{H}{ x^{(k)} }
=
\D{G}{ x^{(k)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(k)} }
=
\D{G}{ x^{(k)} } + \D{G}{ z^{(j)} } \frac{k}{j} y^{(j-k)}
For
k = 1 , \ldots , j
The partial of
H
with respect to
y^{j-k}
,
is given by
\D{H}{ y^{(j-k)} }
=
\D{G}{ y^{(j-k)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ y^{(j-k)} }
=
\D{G}{ y^{(j-k)} } + \D{G}{ z^{(j)} } \frac{k}{j} x^{k}
Order Zero Z(t)
The
z^{(0)}
coefficient
is expressed as a function of the Taylor coefficients
for
X(t)
and
Y(t)
as follows:
In this case,
\D{H}{ x^{(0)} }
=
\D{G}{ x^{(0)} } + \D{G}{ z^{(0)} } \D{ z^{(0)} }{ x^{(0)} }
=
\D{G}{ x^{(0)} } + \D{G}{ z^{(0)} } y^{(0)}
Input File: omh/theory/erf_reverse.omh
