powseries - Maple Help

Representation of Formal Power Series

 Calling Sequence function(arguments)

Description

 • Formal power series are procedures that return the coefficients of the power series they represent. Thus $\mathrm{name}\left(i\right)$ is the coefficient of ${x}^{i}$ in the series called name.
 • The values of the coefficients are saved using a remember table, so all computed coefficients can be seen via op(4, op(name)).
 • The actual procedure is identical for all power series; the only differences are the general term and the values that each remembers.
 • The general term of the power series can be obtained via name(_k) .
 • Note: that each intermediate power series created in a calculation should be named.

Examples

 > $\mathrm{with}\left(\mathrm{powseries}\right):$
 > $\mathrm{powcreate}\left(e\left(n\right)=\frac{1}{n!},e\left(0\right)=1\right):$
 > $\mathrm{powcreate}\left(f\left(n\right)=\frac{f\left(n-1\right)}{{n}^{2}f\left(n-2\right)},f\left(0\right)=1,f\left(1\right)=5,f\left(2\right)=2\right):$
 > $\mathrm{tpsform}\left(e,x,6\right)$
 ${1}{+}{x}{+}\frac{{1}}{{2}}{}{{x}}^{{2}}{+}\frac{{1}}{{6}}{}{{x}}^{{3}}{+}\frac{{1}}{{24}}{}{{x}}^{{4}}{+}\frac{{1}}{{120}}{}{{x}}^{{5}}{+}{O}{}\left({{x}}^{{6}}\right)$ (1)
 > $\mathrm{tpsform}\left(f,x,6\right)$
 ${1}{+}{5}{}{x}{+}{2}{}{{x}}^{{2}}{+}\frac{{2}}{{45}}{}{{x}}^{{3}}{+}\frac{{1}}{{720}}{}{{x}}^{{4}}{+}\frac{{1}}{{800}}{}{{x}}^{{5}}{+}{O}{}\left({{x}}^{{6}}\right)$ (2)

Compute the series  f^e .

 > $\mathrm{logf}≔\mathrm{powlog}\left(f\right):$
 > $\mathrm{elogf}≔\mathrm{multiply}\left(e,\mathrm{logf}\right):$
 > $\mathrm{result}≔\mathrm{powexp}\left(\mathrm{elogf}\right):$
 > $\mathrm{tpsform}\left(\mathrm{result},x,6\right)$
 ${1}{+}{5}{}{x}{+}{7}{}{{x}}^{{2}}{+}\frac{{767}}{{45}}{}{{x}}^{{3}}{+}\frac{{2351}}{{240}}{}{{x}}^{{4}}{+}\frac{{39231}}{{800}}{}{{x}}^{{5}}{+}{O}{}\left({{x}}^{{6}}\right)$ (3)