powsin - Maple Help

powseries

 powsin
 compute the formal power series equal to the sine of an expression

 Calling Sequence powsin(p)

Parameters

 p - formal power series, polynomial, or any function that is acceptable for power series package

Description

 • The function powsin(p) returns the formal power series that is equivalent to $\mathrm{sin}\left(p\right)$.
 • The powseries function evalpow accepts the form as any of powsin, Sin, or sin.
 • The command with(powseries,powsin) allows the use of the abbreviated form of this command.

Examples

 > $\mathrm{with}\left(\mathrm{powseries}\right):$
 > $a≔\mathrm{powsin}\left({x}^{2}\right):$
 > $b≔\mathrm{tpsform}\left(a,x,10\right)$
 ${b}{≔}{{x}}^{{2}}{-}\frac{{1}}{{6}}{}{{x}}^{{6}}{+}{O}{}\left({{x}}^{{10}}\right)$ (1)
 > $c≔\mathrm{evalpow}\left(\mathrm{sin}\left(\mathrm{exp}\left(x\right)\right)\right):$
 > $d≔\mathrm{tpsform}\left(c,x,5\right)$
 ${d}{≔}{\mathrm{sin}}{}\left({1}\right){+}{\mathrm{cos}}{}\left({1}\right){}{x}{+}\left({-}\frac{{\mathrm{sin}}{}\left({1}\right)}{{2}}{+}\frac{{\mathrm{cos}}{}\left({1}\right)}{{2}}\right){}{{x}}^{{2}}{-}\frac{{1}}{{2}}{}{\mathrm{sin}}{}\left({1}\right){}{{x}}^{{3}}{+}\left({-}\frac{{5}{}{\mathrm{cos}}{}\left({1}\right)}{{24}}{-}\frac{{\mathrm{sin}}{}\left({1}\right)}{{4}}\right){}{{x}}^{{4}}{+}{O}{}\left({{x}}^{{5}}\right)$ (2)
 > $e≔\mathrm{powdiff}\left(\mathrm{powsin}\left(x\right)\right):$
 > $f≔\mathrm{tpsform}\left(e,x,6\right)$
 ${f}{≔}{1}{-}\frac{{1}}{{2}}{}{{x}}^{{2}}{+}\frac{{1}}{{24}}{}{{x}}^{{4}}{+}{O}{}\left({{x}}^{{6}}\right)$ (3)