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powseries

 reversion
 reversion of formal power series

 Calling Sequence reversion(a, b)

Parameters

 a - formal power series b - (optional) formal power series

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{powseries}\right):$
 > $\mathrm{powcreate}\left(t\left(n\right)=\frac{t\left(n-1\right)}{n},t\left(0\right)=0,t\left(1\right)=1\right):$
 > $\mathrm{powcreate}\left(v\left(n\right)=\frac{v\left(n-1\right)}{2},v\left(0\right)=0,v\left(1\right)=1\right):$
 > $s≔\mathrm{reversion}\left(t,v\right):$
 > $\mathrm{tpsform}\left(s,x,11\right)$
 ${x}{+}\frac{{1}}{{12}}{}{{x}}^{{3}}{+}\frac{{1}}{{80}}{}{{x}}^{{5}}{+}\frac{{1}}{{448}}{}{{x}}^{{7}}{+}\frac{{1}}{{2304}}{}{{x}}^{{9}}{+}{\mathrm{O}}\left({{x}}^{{11}}\right)$ (1)
 > $\mathrm{ts}≔\mathrm{compose}\left(t,s\right):$
 > $\mathrm{tpsform}\left(\mathrm{ts},x\right)$
 ${x}{+}\frac{{1}}{{2}}{}{{x}}^{{2}}{+}\frac{{1}}{{4}}{}{{x}}^{{3}}{+}\frac{{1}}{{8}}{}{{x}}^{{4}}{+}\frac{{1}}{{16}}{}{{x}}^{{5}}{+}{\mathrm{O}}\left({{x}}^{{6}}\right)$ (2)
 > $\mathrm{tpsform}\left(v,x\right)$
 ${x}{+}\frac{{1}}{{2}}{}{{x}}^{{2}}{+}\frac{{1}}{{4}}{}{{x}}^{{3}}{+}\frac{{1}}{{8}}{}{{x}}^{{4}}{+}\frac{{1}}{{16}}{}{{x}}^{{5}}{+}{\mathrm{O}}\left({{x}}^{{6}}\right)$ (3)