= Inverse of a formal power series
{title2=$f(x)^{-1}$}
= Formal power series inverse
{synonym}
Here an <inverse of a formal power series> means its multiplicative inverse: $f(x)^{-1}f(x)=1$. The <formal power series reciprocal> exists exactly when the constant coefficient of $f$ is a <unit> in the coefficient <ring>. Thus $1-x$ is invertible in $\mathbb Z[[x]]$, while $2+x$ is not, though it is invertible in $\mathbb Q[[x]]$.
A compositional inverse solves $f(g(x))=x$, which is a different problem. For instance, the multiplicative inverse of $1-x$ is $1+x+x^2+\cdots$, whereas the compositional inverse of $x+x^2$ starts $x-x^2+2x^3-\cdots$. Stating which operation is inverted prevents a common ambiguity.
Back to article page
