 type/SERIES - Help

type/SERIES

SERIES data structure

 Calling Sequence type(expr, SERIES)

Parameters

 expr - algebraic expression

Description

 • The function type/SERIES returns true if the value of expr is Maple's SERIES data structure, explained below.
 • The SERIES data structure represents an expression as a truncated series in one specified indeterminate, expanded at a particular point. A call to the MultiSeries[multiseries] function will always return an object of this type, or 0.
 • The SERIES structure has nine entries:
 1 the scale (a table, see MultiSeries,scale);
 2 the list of coefficients, which can be SERIES themselves;
 3 the coefficient of the $\mathrm{O}\left(...\right)$ term, which can be a function of the variable varying more slowly than the expansion variable. It is 0 if the series is exact with respect to its expansion variable (see below);
 4 the common type of its coefficients;
 5 the list of exponents;
 6 the exponent of the $\mathrm{O}\left(...\right)$ term;
 7 the type of the exponents;
 8 the expansion variable, i.e. the element of the asymptotic basis being used;
 9 the expression being expanded.

Examples

 > $\mathrm{with}\left(\mathrm{MultiSeries},\mathrm{multiseries}\right):$
 > $s≔\mathrm{multiseries}\left(\mathrm{sin}\left(x\right),x,3\right)$
 ${s}{≔}{x}{-}\frac{{{x}}^{{3}}}{{6}}{+}{\mathrm{O}}{}\left({{x}}^{{4}}\right)$ (1)
 > $\mathrm{lprint}\left(s\right)$
 SERIES(Scale,[1, -1/6],1,rational,[1, 3],4,integer,_var[x],sin(_var[x]))
 > $s≔\mathrm{multiseries}\left(\frac{1}{x}+5+6{x}^{2},x,3\right)$
 ${s}{≔}\frac{{1}}{{x}}{+}{5}{+}{6}{}{{x}}^{{2}}$ (2)
 > $\mathrm{lprint}\left(s\right)$
 SERIES(Scale,[1, 5, 6],0,integer,[-1, 0, 2],infinity,integer,_var[x],1/_var[x]+5+6*_var[x]^2)
 > $f≔\frac{1}{1-\frac{{ⅇ}^{-\frac{1}{x}}}{1-x}}-1$
 ${f}{≔}\frac{{1}}{{1}{-}\frac{{{ⅇ}}^{{-}\frac{{1}}{{x}}}}{{1}{-}{x}}}{-}{1}$ (3)
 > $s≔\mathrm{multiseries}\left(f,x,1\right):$
 > $s≔\mathrm{multiseries}\left(f,\mathrm{op}\left(1,s\right),3,{{\mathrm{op}\left(1,s\right)}_{\mathrm{list}}}_{2..2}\right)$
 ${s}{≔}\frac{{1}}{\left({1}{-}{x}\right){}{{ⅇ}}^{\frac{{1}}{{x}}}}{+}\frac{{1}}{{\left({1}{-}{x}\right)}^{{2}}{}{\left({{ⅇ}}^{\frac{{1}}{{x}}}\right)}^{{2}}}{+}{\mathrm{O}}{}\left(\frac{{1}}{{\left({{ⅇ}}^{\frac{{1}}{{x}}}\right)}^{{3}}}\right)$ (4)
 > $\mathrm{lprint}\left(s\right)$
 SERIES(Scale,[1/(1-_var[x]), 1/(1-_var[x])^2],1,algebraic,[1, 2],3,integer,_var[1/exp(1/x)],-1+1/(1-_var[1/exp(1/x)]/(1-_var[x])))