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MultiSeries

 SeriesInfo
 return information about a multiseries

 Calling Sequence SeriesInfo:-ListCoefficients(S) SeriesInfo:-ListExponents(S) SeriesInfo:-CoefficientBigO(S) SeriesInfo:-ExponentBigO(S) SeriesInfo:-Variable(S) SeriesInfo:-Scale(S)

Parameters

 S - a multiseries

Description

 • The SeriesInfo subpackage is a collection of simple procedures that return information about multiseries. They are intended to serve as a programmer interface to the SERIES data structure. For compatibility with future releases, it is strongly recommended that you use these commands instead of accessing the operands of the data structure directly.
 • The ListCoefficients function returns the list of coefficients of S.
 • The ListExponents function returns the list of exponents of S.
 • The CoefficientBigO function returns the coefficient in the $\mathrm{O}\left(...\right)$ term of S.
 • The ExponentBigO function returns the exponent of the variable in the $\mathrm{O}\left(...\right)$ term of S.
 • A multiseries S without an $\mathrm{O}\left(...\right)$ term yields $\mathrm{CoefficientBigO}\left(S\right)=0$ and $\mathrm{ExponentBigO}\left(S\right)=\mathrm{\infty }$.
 • The Variable function returns the variable used in the series S.
 • The Scale function returns the asymptotic scale in which S has been expanded.

Examples

 > $\mathrm{with}\left(\mathrm{MultiSeries}\right):$
 > $\mathrm{with}\left(\mathrm{SeriesInfo}\right):$

An example with a simple power series expansion:

 > $s≔\mathrm{multiseries}\left(\mathrm{sin}\left(x\right),x\right)$
 ${s}{≔}{x}{-}\frac{{{x}}^{{3}}}{{6}}{+}\frac{{{x}}^{{5}}}{{120}}{+}{\mathrm{O}}{}\left({{x}}^{{7}}\right)$ (1)
 > $\mathrm{ListCoefficients}\left(s\right)$
 $\left[{1}{,}{-}\frac{{1}}{{6}}{,}\frac{{1}}{{120}}\right]$ (2)
 > $\mathrm{ListExponents}\left(s\right)$
 $\left[{1}{,}{3}{,}{5}\right]$ (3)
 > $\mathrm{CoefficientBigO}\left(s\right)$
 ${1}$ (4)
 > $\mathrm{ExponentBigO}\left(s\right)$
 ${7}$ (5)
 > $\mathrm{Variable}\left(s\right)$
 ${x}$ (6)

A more complicated expansion:

 > $S≔\mathrm{multiseries}\left(\mathrm{GAMMA}\left(x\right),x=\mathrm{∞}\right)$
 ${S}{≔}\left(\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}{}\sqrt{\frac{{1}}{{x}}}{+}\frac{\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}{}{\left(\frac{{1}}{{x}}\right)}^{{3}}{{2}}}}{{12}}{+}\frac{\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}{}{\left(\frac{{1}}{{x}}\right)}^{{5}}{{2}}}}{{288}}{-}\frac{{139}{}\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}{}{\left(\frac{{1}}{{x}}\right)}^{{7}}{{2}}}}{{51840}}{-}\frac{{571}{}\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}{}{\left(\frac{{1}}{{x}}\right)}^{{9}}{{2}}}}{{2488320}}{+}\frac{{163879}{}\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}{}{\left(\frac{{1}}{{x}}\right)}^{{11}}{{2}}}}{{209018880}}{+}{\mathrm{O}}{}\left({\left(\frac{{1}}{{x}}\right)}^{{13}}{{2}}}\right)\right){}{{ⅇ}}^{\left({\mathrm{ln}}{}\left({x}\right){-}{1}\right){}{x}}$ (7)
 > $\mathrm{Variable}\left(S\right)$
 $\frac{{1}}{{{ⅇ}}^{\left({\mathrm{ln}}{}\left({x}\right){-}{1}\right){}{x}}}$ (8)
 > $\mathrm{CoefficientBigO}\left(S\right)$
 ${0}$ (9)
 > $\mathrm{ExponentBigO}\left(S\right)$
 ${\mathrm{\infty }}$ (10)
 > $\mathrm{listcoeff}≔\mathrm{ListCoefficients}\left(S\right)$
 ${\mathrm{listcoeff}}{≔}\left[\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}{}\sqrt{\frac{{1}}{{x}}}{+}\frac{\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}{}{\left(\frac{{1}}{{x}}\right)}^{{3}}{{2}}}}{{12}}{+}\frac{\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}{}{\left(\frac{{1}}{{x}}\right)}^{{5}}{{2}}}}{{288}}{-}\frac{{139}{}\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}{}{\left(\frac{{1}}{{x}}\right)}^{{7}}{{2}}}}{{51840}}{-}\frac{{571}{}\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}{}{\left(\frac{{1}}{{x}}\right)}^{{9}}{{2}}}}{{2488320}}{+}\frac{{163879}{}\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}{}{\left(\frac{{1}}{{x}}\right)}^{{11}}{{2}}}}{{209018880}}{+}{\mathrm{O}}{}\left({\left(\frac{{1}}{{x}}\right)}^{{13}}{{2}}}\right)\right]$ (11)
 > $L≔{\mathrm{listcoeff}}_{1}$
 ${L}{≔}\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}{}\sqrt{\frac{{1}}{{x}}}{+}\frac{\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}{}{\left(\frac{{1}}{{x}}\right)}^{{3}}{{2}}}}{{12}}{+}\frac{\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}{}{\left(\frac{{1}}{{x}}\right)}^{{5}}{{2}}}}{{288}}{-}\frac{{139}{}\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}{}{\left(\frac{{1}}{{x}}\right)}^{{7}}{{2}}}}{{51840}}{-}\frac{{571}{}\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}{}{\left(\frac{{1}}{{x}}\right)}^{{9}}{{2}}}}{{2488320}}{+}\frac{{163879}{}\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}{}{\left(\frac{{1}}{{x}}\right)}^{{11}}{{2}}}}{{209018880}}{+}{\mathrm{O}}{}\left({\left(\frac{{1}}{{x}}\right)}^{{13}}{{2}}}\right)$ (12)
 > $\mathrm{type}\left(L,\mathrm{SERIES}\right)$
 ${\mathrm{true}}$ (13)
 > $\mathrm{Variable}\left(L\right)$
 $\frac{{1}}{{x}}$ (14)
 > $\mathrm{ListCoefficients}\left(L\right)$
 $\left[\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}{,}\frac{\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}}{{12}}{,}\frac{\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}}{{288}}{,}{-}\frac{{139}{}\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}}{{51840}}{,}{-}\frac{{571}{}\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}}{{2488320}}{,}\frac{{163879}{}\sqrt{{2}}{}\sqrt{{\mathrm{\pi }}}}{{209018880}}\right]$ (15)
 > $\mathrm{ListExponents}\left(L\right)$
 $\left[\frac{{1}}{{2}}{,}\frac{{3}}{{2}}{,}\frac{{5}}{{2}}{,}\frac{{7}}{{2}}{,}\frac{{9}}{{2}}{,}\frac{{11}}{{2}}\right]$ (16)

 See Also