= Cubic series reversion
{title2=$g=f/a_1-a_2f^2/a_1^3+(2a_2^2-a_1a_3)f^3/a_1^5+O(f^4)$}
Suppose $f=a_1g+a_2g^2+a_3g^3+O(g^4)$ with $a_1\ne0$. Substitute $g=b_1f+b_2f^2+b_3f^3+O(f^4)$ and compare coefficients. The three equations are $a_1b_1=1$, $a_1b_2+a_2b_1^2=0$, and $a_1b_3+2a_2b_1b_2+a_3b_1^3=0$. Solving them proves the title formula. This is <series reversion> for a local <inverse function>; analyticity guarantees a convergent inverse series sufficiently near the origin.
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