asympt - Maple Help

asympt

asymptotic expansion

 Calling Sequence asympt(f, x) asympt(f, x, n)

Parameters

 f - algebraic expression in x x - name n - positive integer (truncation order)

Description

 • The function asympt computes the asymptotic expansion of f with respect to the variable x (as x approaches infinity).
 • The third argument n specifies the truncation order of the series expansion.  If no third argument is given, the value of the global variable Order (default Order = 6) is used.
 • Specifically, asympt is defined in terms of the series function

$\mathrm{subs}\left(x=\frac{1}{x},\mathrm{series}\left(\mathrm{subs}\left(x=\frac{1}{x},f\right),x=0,n\right)\right)$

 However, the expression returned will be in sum-of-products form rather than in the series form.

Examples

 > $\mathrm{asympt}\left(\frac{x}{1-x-{x}^{2}},x\right)$
 ${-}\frac{{1}}{{x}}{+}\frac{{1}}{{{x}}^{{2}}}{-}\frac{{2}}{{{x}}^{{3}}}{+}\frac{{3}}{{{x}}^{{4}}}{-}\frac{{5}}{{{x}}^{{5}}}{+}{\mathrm{O}}{}\left(\frac{{1}}{{{x}}^{{6}}}\right)$ (1)
 > $\mathrm{asympt}\left(\frac{n!}{\mathrm{sqrt}\left(2\mathrm{\pi }\right)},n,3\right)$
 $\frac{\frac{{1}}{\sqrt{\frac{{1}}{{n}}}}{+}\frac{\sqrt{\frac{{1}}{{n}}}}{{12}}{+}\frac{{\left(\frac{{1}}{{n}}\right)}^{{3}}{{2}}}}{{288}}{+}{\mathrm{O}}{}\left({\left(\frac{{1}}{{n}}\right)}^{{5}}{{2}}}\right)}{{\left(\frac{{1}}{{n}}\right)}^{{n}}{}{{ⅇ}}^{{n}}}$ (2)
 > $\mathrm{asympt}\left(\mathrm{exp}\left({x}^{2}\right)\left(1-\mathrm{erf}\left(x\right)\right),x\right)$
 $\frac{{1}}{\sqrt{{\mathrm{\pi }}}{}{x}}{-}\frac{{1}}{{2}{}\sqrt{{\mathrm{\pi }}}{}{{x}}^{{3}}}{+}\frac{{3}}{{4}{}\sqrt{{\mathrm{\pi }}}{}{{x}}^{{5}}}{+}{\mathrm{O}}{}\left(\frac{{1}}{{{x}}^{{7}}}\right)$ (3)
 > $\mathrm{asympt}\left(\mathrm{sqrt}\left(\frac{\mathrm{\pi }}{2}\right)\mathrm{BesselJ}\left(0,x\right),x,3\right)$
 ${\mathrm{sin}}{}\left({x}{+}\frac{{\mathrm{\pi }}}{{4}}\right){}\sqrt{\frac{{1}}{{x}}}{-}\frac{{\mathrm{cos}}{}\left({x}{+}\frac{{\mathrm{\pi }}}{{4}}\right){}{\left(\frac{{1}}{{x}}\right)}^{{3}}{{2}}}}{{8}}{-}\frac{{9}{}{\mathrm{sin}}{}\left({x}{+}\frac{{\mathrm{\pi }}}{{4}}\right){}{\left(\frac{{1}}{{x}}\right)}^{{5}}{{2}}}}{{128}}{+}{\mathrm{O}}{}\left({\left(\frac{{1}}{{x}}\right)}^{{7}}{{2}}}\right)$ (4)
 > $\mathrm{series}\left(\frac{\mathrm{ln}\left(x\right)}{x-1},x,8\right)$
 ${-}{\mathrm{ln}}{}\left({x}\right){-}{\mathrm{ln}}{}\left({x}\right){}{x}{-}{\mathrm{ln}}{}\left({x}\right){}{{x}}^{{2}}{-}{\mathrm{ln}}{}\left({x}\right){}{{x}}^{{3}}{-}{\mathrm{ln}}{}\left({x}\right){}{{x}}^{{4}}{-}{\mathrm{ln}}{}\left({x}\right){}{{x}}^{{5}}{-}{\mathrm{ln}}{}\left({x}\right){}{{x}}^{{6}}{-}{\mathrm{ln}}{}\left({x}\right){}{{x}}^{{7}}{+}{O}{}\left({{x}}^{{8}}\right)$ (5)
 > $\mathrm{asympt}\left(\frac{\mathrm{ln}\left(x\right)}{x-1},x,8\right)$
 $\frac{{\mathrm{ln}}{}\left({x}\right)}{{x}}{+}\frac{{\mathrm{ln}}{}\left({x}\right)}{{{x}}^{{2}}}{+}\frac{{\mathrm{ln}}{}\left({x}\right)}{{{x}}^{{3}}}{+}\frac{{\mathrm{ln}}{}\left({x}\right)}{{{x}}^{{4}}}{+}\frac{{\mathrm{ln}}{}\left({x}\right)}{{{x}}^{{5}}}{+}\frac{{\mathrm{ln}}{}\left({x}\right)}{{{x}}^{{6}}}{+}\frac{{\mathrm{ln}}{}\left({x}\right)}{{{x}}^{{7}}}{+}{\mathrm{O}}{}\left(\frac{{1}}{{{x}}^{{8}}}\right)$ (6)

Compatibility

 • The asympt command was updated in Maple 2016; see Advanced Math.