The function reversion(a, b) returns the formal power series that is the reversion of a with respect to b.  If b is not specified then it is assumed to be the formal power series with one nonzero coefficient, $b\left(1\right)=1$.

Since reversion is the inverse of composition, composition of the result into a will give b.

Note that $a\left(0\right)$ must be $0$, $a\left(1\right)$ must be $1$, and $b\left(0\right)$ must be $0$. If not, the reversion is not well defined and reversion returns an error message.

The command with(powseries,reversion) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{powseries}\right)\:$

$\mathrm{powcreate}\left(t\left(n\right)=\frac{t\left(n-1\right)}{n}\,t\left(0\right)=0\,t\left(1\right)=1\right)\:$

$\mathrm{powcreate}\left(v\left(n\right)=\frac{v\left(n-1\right)}{2}\,v\left(0\right)=0\,v\left(1\right)=1\right)\:$

$s≔\mathrm{reversion}\left(t\,v\right)\:$

$\mathrm{tpsform}\left(s\,x\,11\right)$

$x+\frac{1}{12}{x}^3+\frac{1}{80}{x}^5+\frac{1}{448}{x}^7+\frac{1}{2304}{x}^9+O\left(x^{11}\right)$

$\mathrm{ts}≔\mathrm{compose}\left(t\,s\right)\:$

$\mathrm{tpsform}\left(\mathrm{ts}\,x\right)$

$x+\frac{1}{2}{x}^2+\frac{1}{4}{x}^3+\frac{1}{8}{x}^4+\frac{1}{16}{x}^5+O\left(x^6\right)$

$\mathrm{tpsform}\left(v\,x\right)$

$x+\frac{1}{2}{x}^2+\frac{1}{4}{x}^3+\frac{1}{8}{x}^4+\frac{1}{16}{x}^5+O\left(x^6\right)$