I have some fractional expression like
$$ \frac{x^2+xyz+z^2}{x^2-yz+x^2z} $$
and I know that $x \ll y \ll z$. Really I want to divide through by $z^2$ and then take a Taylor series expansion in terms of $\frac{x}{z}$ and $\frac{y}{z}$.
I have tried this approach:
- Divide the numerator alone by $z^2$ using
Numerator
- Replace terms like $\frac{x}{z} \to \frac{x}{z} \epsilon$
- Then I would take a Taylor series in $\epsilon$ and let $\epsilon \to 1$. (or maybe just set terms like $\epsilon^2 \to 0$)
I have two questions:
Is there a better way than separating the numerator and denominator and treating them separately, and more pressingly
ReplaceAll
(/.
) won't work on a compound expression like $\frac{x y}{z^2}$, I want $\frac{x y}{z^2} \to \frac{x y}{z^2} \epsilon^2$ but the replacement doesn't work on these compound expressions
Thanks
EDIT: sorry about the fraction above, it won't let me post between $$ $$
symbols
x
andy
by the same quantities multiplied byϵ
and useSeries
for the expansion inϵ
. $\endgroup$