The function asympt computes the asymptotic expansion of f with respect to the variable x (as x approaches infinity).

The asympt function of the MultiSeries package is intended to be used in the same manner as the top-level asympt function.

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.

Like the asympt function, the result is returned in sum-of-products form.

The underlying engine for computing expansions is the MultiSeries[multiseries] function. In particular, the variable x is assumed to tend to infinity along the positive real axis (this can be modified by a change of variables).

In rare cases, it might be necessary to increase the value of the global variable Order in order to improve the ability of asympt to solve problems with significant cancellation. This is made explicit by an error message coming from multiseries.

It can also happen that the result is wrong because Testzero failed to recognize that the leading coefficient of a multiseries expansion happens to be 0. In those cases, it is necessary to modify this environment variable (see Testzero).

The call asympt(f,x) is equivalent to series(f,x=infinity).

$\mathrm{with}\left(\mathrm{MultiSeries}\,\mathrm{asympt}\right)\:$

$\mathrm{asympt}\left(\frac{x}{1-x-x^2}\,x\right)$

$-\frac{5}{x^5}+\frac{3}{x^4}-\frac{2}{x^3}+\frac{1}{x^2}-\frac{1}{x}+\mathrm{O}\left(\frac{1}{x^6}\right)$

$\mathrm{asympt}\left(\frac{n!\exp \left(n\right)}{n^n\mathrm{sqrt}\left(2\mathrm{\pi }\right)}\,n\,3\right)$

$\frac{1}{\sqrt{\frac{1}{n}}}+\frac{\sqrt{\frac{1}{n}}}{12}+\frac{{\left(\frac{1}{n}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{288}+\mathrm{O}\left({\left(\frac{1}{n}\right)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)$

$\mathrm{asympt}\left(\mathrm{sqrt}\left(\frac{\mathrm{\pi }}{2}\right)\mathrm{BesselJ}\left(0\,x\right)\,x\,3\right)$

$\left(\frac{\sqrt{2}\sin \left(x\right)}{2}+\frac{\sqrt{2}\cos \left(x\right)}{2}\right)\sqrt{\frac{1}{x}}+\left(-\frac{\sqrt{2}\cos \left(x\right)}{16}+\frac{\sqrt{2}\sin \left(x\right)}{16}\right){\left(\frac{1}{x}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+\left(-\frac{9\sqrt{2}\sin \left(x\right)}{256}-\frac{9\sqrt{2}\cos \left(x\right)}{256}\right){\left(\frac{1}{x}\right)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+\mathrm{O}\left(\left(-\frac{75\mathrm{I}\sqrt{2}\left(\frac{\sqrt{2}}{2}+\frac{\mathrm{I}\sqrt{2}}{2}\right){\ⅇ}^{\mathrm{-I}x}}{2048\sqrt{\mathrm{\pi }}}+\frac{75\mathrm{I}\sqrt{2}\left(\frac{\sqrt{2}}{2}-\frac{\mathrm{I}\sqrt{2}}{2}\right){\ⅇ}^{\mathrm{I}x}}{2048\sqrt{\mathrm{\pi }}}\right){\left(\frac{1}{x}\right)}^{\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)$