Online Help

The function compose(a, b) computes and returns the formal power series $a\left(b\right)$, the composition of a and b. This power series is: $a\left(0\right)+a\left(1\right)b+a\left(2\right)b^2+a\left(3\right)b^3+...$.

Note that $b\left(0\right)$ must be 0 to ensure that the composition is well defined. Otherwise, compose returns an error message.

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

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

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

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

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

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

$1+x+x^2+\frac{11}{12}{x}^3+\frac{19}{24}{x}^4+\frac{157}{240}{x}^5+O\left(x^6\right)$