Code: Select all
\begin{align*}
w &=& f(z,w) \\
&=& z^3 + (a_1z + a_2z^2)w + (a_3 + a_4z)w^2 + a_6w^3 \\
&=& z^3 + (a_1z + a_2z^2)f(z,w) + (a_3 + a_4z)f(z,w)^2 + a_6f(z,w)^3 \\
&=& z^3 + (a_1z + a_2z^2)(z^3 + (a_1z + a_2z^2)w + (a_3 + a_4z)w^2 + a_6w^3) + \\
& & (a_3 + a_4z)(z^3 + (a_1z + a_2z^2)w + (a_3 + a_4z)w^2 + a_6w^3)^2 + a_6(z^3 + \\
& & (a_1z + a_2z^2)w + (a_3 + a_4z)w^2 + a_6w^3)^3 \\
&=& \ldots \\
&=& z^3 + a_1z^4 + (a_1^2 + a_2)z^5 + (a_1^3 + 2a_1a_2 + a_3)z^6 + \\
& & (a_1^4 + 3a_1^2a_2 + 3a_1a_3 + a_2^2 + a_4)z^7 + \cdots \\
&=& z^3(1 + A_1z + A_2z^2 + \cdots)
\end{align*}
Any help would be appreciated, thank you.