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The MultiSeries package

 Calling Sequence MultiSeries:-command(arguments) command(arguments)

Description

 • The MultiSeries package is still under development.  It is very likely that the functionality will change in the next release of Maple.  Thus code using this package may not be backwards compatible with the next release of Maple.
 • The MultiSeries package performs asymptotic and series expansions in general asymptotic scales.
 • The functions provided by the package are:

 • For help with a particular function see MultiSeries:-command, where command is taken from the above list.
 • The simplest use of the package is by overriding the standard asympt, series, limit using with. The corresponding MultiSeries functions are often more powerful than the default ones, but require more computational time.
 • The variable is assumed to tend to its limit along a ray coming from the origin. When the limit point is 0, the variable tends to $0+$. This direction can be changed by a simple change of variable (see the examples in multiseries).
 • Two other differences with the original functions are that the $\mathrm{O}\left(...\right)$ term corresponds more closely to the classical mathematical definition, and the result is undefined instead of a range for limits of oscillating functions.
 • The most comprehensive access to multiseries expansion in the MultiSeries package is provided by the function multiseries.

Examples

Computations that cannot be performed with the standard limit, series, asympt:

 > $\mathrm{limit}\left(\frac{\mathrm{ln}\left(\frac{\mathrm{\Gamma }\left(Ix\right)}{\mathrm{\Gamma }\left(x\right)}\right)}{x\mathrm{ln}\left(x\right)},x=\mathrm{\infty }\right)$
 $\underset{{x}{\to }{\mathrm{\infty }}}{{lim}}{}\frac{{\mathrm{ln}}{}\left(\frac{{\mathrm{\Gamma }}{}\left({I}{}{x}\right)}{{\mathrm{\Gamma }}{}\left({x}\right)}\right)}{{x}{}{\mathrm{ln}}{}\left({x}\right)}$ (1)
 > $\mathrm{asympt}\left(\mathrm{binomial}\left(n+a,n\right),n,3\right)$
 > $\mathrm{asympt}\left(\frac{\mathrm{\Gamma }\left(x+\mathrm{exp}\left(-x\right)\right)}{\mathrm{\Gamma }\left(x\right)}-1,x,3\right)$
 ${\mathrm{O}}{}\left(\frac{{1}}{{{x}}^{{3}}}\right)$ (2)
 > $\mathrm{series}\left(\mathrm{BesselJ}\left(a,z\right),z,3\right)$

Computations that lead to a different result:

 > $\mathrm{series}\left(\frac{1}{1-x\mathrm{ln}\left(x\right)},x,3\right)$
 ${1}{+}{\mathrm{ln}}{}\left({x}\right){}{x}{+}{{\mathrm{ln}}{}\left({x}\right)}^{{2}}{}{{x}}^{{2}}{+}{O}{}\left({{x}}^{{3}}\right)$ (3)
 > $\mathrm{limit}\left(\mathrm{sin}\left(x\right),x=\mathrm{\infty }\right)$
 ${-1}{..}{1}$ (4)

The same commands after loading the package:

 > $\mathrm{with}\left(\mathrm{MultiSeries}\right)$
 $\left[{\mathrm{AddFunction}}{,}{\mathrm{FunctionSupported}}{,}{\mathrm{GetFunction}}{,}{\mathrm{LeadingTerm}}{,}{\mathrm{RemoveFunction}}{,}{\mathrm{SeriesInfo}}{,}{\mathrm{asympt}}{,}{\mathrm{limit}}{,}{\mathrm{multiseries}}{,}{\mathrm{series}}{,}{\mathrm{taylor}}\right]$ (5)
 > $\mathrm{limit}\left(\frac{\mathrm{ln}\left(\frac{\mathrm{\Gamma }\left(Ix\right)}{\mathrm{\Gamma }\left(x\right)}\right)}{x\mathrm{ln}\left(x\right)},x=\mathrm{\infty }\right)$
 ${-1}$ (6)
 > $\mathrm{asympt}\left(\mathrm{binomial}\left(n+a,n\right),n,3\right)$
 $\left(\frac{{\mathrm{cos}}{}\left({\mathrm{ln}}{}\left({n}\right){}{\mathrm{\Im }}{}\left({a}\right)\right)}{{\mathrm{\Gamma }}{}\left({a}\right){}{a}}{+}\frac{{I}{}{\mathrm{sin}}{}\left({\mathrm{ln}}{}\left({n}\right){}{\mathrm{\Im }}{}\left({a}\right)\right)}{{\mathrm{\Gamma }}{}\left({a}\right){}{a}}\right){}{\left(\frac{{1}}{{n}}\right)}^{{-}{\mathrm{\Re }}{}\left({a}\right)}{+}\left(\frac{{\mathrm{cos}}{}\left({\mathrm{ln}}{}\left({n}\right){}{\mathrm{\Im }}{}\left({a}\right)\right){}{a}}{{2}{}{\mathrm{\Gamma }}{}\left({a}\right)}{+}\frac{{\mathrm{cos}}{}\left({\mathrm{ln}}{}\left({n}\right){}{\mathrm{\Im }}{}\left({a}\right)\right)}{{2}{}{\mathrm{\Gamma }}{}\left({a}\right)}{+}\frac{{I}{}{\mathrm{sin}}{}\left({\mathrm{ln}}{}\left({n}\right){}{\mathrm{\Im }}{}\left({a}\right)\right){}{a}}{{2}{}{\mathrm{\Gamma }}{}\left({a}\right)}{+}\frac{{I}{}{\mathrm{sin}}{}\left({\mathrm{ln}}{}\left({n}\right){}{\mathrm{\Im }}{}\left({a}\right)\right)}{{2}{}{\mathrm{\Gamma }}{}\left({a}\right)}\right){}{\left(\frac{{1}}{{n}}\right)}^{{1}{-}{\mathrm{\Re }}{}\left({a}\right)}{+}\left(\frac{{\mathrm{cos}}{}\left({\mathrm{ln}}{}\left({n}\right){}{\mathrm{\Im }}{}\left({a}\right)\right){}{{a}}^{{3}}}{{8}{}{\mathrm{\Gamma }}{}\left({a}\right)}{+}\frac{{\mathrm{cos}}{}\left({\mathrm{ln}}{}\left({n}\right){}{\mathrm{\Im }}{}\left({a}\right)\right){}{{a}}^{{2}}}{{12}{}{\mathrm{\Gamma }}{}\left({a}\right)}{-}\frac{{\mathrm{cos}}{}\left({\mathrm{ln}}{}\left({n}\right){}{\mathrm{\Im }}{}\left({a}\right)\right){}{a}}{{8}{}{\mathrm{\Gamma }}{}\left({a}\right)}{-}\frac{{\mathrm{cos}}{}\left({\mathrm{ln}}{}\left({n}\right){}{\mathrm{\Im }}{}\left({a}\right)\right)}{{12}{}{\mathrm{\Gamma }}{}\left({a}\right)}{+}\frac{{I}{}{\mathrm{sin}}{}\left({\mathrm{ln}}{}\left({n}\right){}{\mathrm{\Im }}{}\left({a}\right)\right){}{{a}}^{{3}}}{{8}{}{\mathrm{\Gamma }}{}\left({a}\right)}{+}\frac{{I}{}{\mathrm{sin}}{}\left({\mathrm{ln}}{}\left({n}\right){}{\mathrm{\Im }}{}\left({a}\right)\right){}{{a}}^{{2}}}{{12}{}{\mathrm{\Gamma }}{}\left({a}\right)}{-}\frac{{I}{}{\mathrm{sin}}{}\left({\mathrm{ln}}{}\left({n}\right){}{\mathrm{\Im }}{}\left({a}\right)\right){}{a}}{{8}{}{\mathrm{\Gamma }}{}\left({a}\right)}{-}\frac{{I}{}{\mathrm{sin}}{}\left({\mathrm{ln}}{}\left({n}\right){}{\mathrm{\Im }}{}\left({a}\right)\right)}{{12}{}{\mathrm{\Gamma }}{}\left({a}\right)}\right){}{\left(\frac{{1}}{{n}}\right)}^{{2}{-}{\mathrm{\Re }}{}\left({a}\right)}{+}{\mathrm{O}}{}\left(\left(\left({-}\frac{{\left({a}{+}{1}\right)}^{{2}}}{{576}}{+}\left({a}{+}\frac{{1}}{{2}}\right){}\left(\frac{{a}}{{288}}{+}\frac{{1}}{{288}}\right){+}\frac{{1}}{{3456}}\right){}{{ⅇ}}^{{I}{}{\mathrm{ln}}{}\left({n}\right){}{\mathrm{\Im }}{}\left({a}\right)}{+}{{ⅇ}}^{{I}{}{\mathrm{ln}}{}\left({n}\right){}{\mathrm{\Im }}{}\left({a}\right)}{}\left({-}\frac{{\left({a}{+}{1}\right)}^{{3}}}{{36}}{+}\frac{{\left({a}{+}{1}\right)}^{{2}}{}\left({a}{+}\frac{{1}}{{2}}\right)}{{24}}{+}\frac{{a}}{{144}}{+}\frac{{1}}{{144}}{-}\frac{{\left({-}\frac{{\left({a}{+}{1}\right)}^{{2}}}{{2}}{+}\left({a}{+}{1}\right){}\left({a}{+}\frac{{1}}{{2}}\right){+}\frac{{1}}{{12}}\right)}^{{2}}}{{24}}\right){+}{1}{+}{{ⅇ}}^{{I}{}{\mathrm{ln}}{}\left({n}\right){}{\mathrm{\Im }}{}\left({a}\right)}{}\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}\right){}{\left(\frac{{1}}{{n}}\right)}^{{3}{-}{\mathrm{\Re }}{}\left({a}\right)}\right)$ (7)
 > $\mathrm{asympt}\left(\frac{\mathrm{\Gamma }\left(x+\mathrm{exp}\left(-x\right)\right)}{\mathrm{\Gamma }\left(x\right)}-1,x,3\right)$
 $\frac{{\mathrm{ln}}{}\left({x}\right){-}\frac{{1}}{{2}{}{x}}{-}\frac{{1}}{{12}{}{{x}}^{{2}}}{+}{\mathrm{O}}{}\left(\frac{{\mathrm{ln}}{}\left({x}\right)}{{{x}}^{{3}}}\right)}{{{ⅇ}}^{{x}}}$ (8)
 > $\mathrm{series}\left(\mathrm{BesselJ}\left(a,z\right),z,3\right)$
 $\frac{{{z}}^{{a}}}{{\mathrm{\Gamma }}{}\left({a}{+}{1}\right){}{{ⅇ}}^{{\mathrm{ln}}{}\left({2}\right){}{a}}}{-}\frac{{{z}}^{{2}{+}{a}}}{{4}{}\left({a}{+}{1}\right){}{\mathrm{\Gamma }}{}\left({a}{+}{1}\right){}{{ⅇ}}^{{\mathrm{ln}}{}\left({2}\right){}{a}}}{+}{\mathrm{O}}{}\left({{z}}^{{4}{+}{a}}\right)$ (9)
 > $\mathrm{series}\left(\frac{1}{1-x\mathrm{ln}\left(x\right)},x,3\right)$
 ${1}{-}{\mathrm{ln}}{}\left(\frac{{1}}{{x}}\right){}{x}{+}{{\mathrm{ln}}{}\left(\frac{{1}}{{x}}\right)}^{{2}}{}{{x}}^{{2}}{+}{O}{}\left({{x}}^{{3}}\right)$ (10)
 > $\mathrm{limit}\left(\mathrm{sin}\left(x\right),x=\mathrm{\infty }\right)$
 ${\mathrm{undefined}}$ (11)

The LeadingTerm command is a useful way to find the leading term in the asymptotic or local behavior of a function:

 > $\mathrm{LeadingTerm}\left(\mathrm{ln}\left(\mathrm{cot}\left(\frac{\mathrm{ln}\left(x\right)}{x}\right)\right),x=\mathrm{\infty }\right)$
 ${\mathrm{ln}}{}\left({x}\right)$ (12)

 See Also