The command PowerSeries(p), where p is a polynomial, creates a power series representing this polynomial. Concretely, the homogeneous part of degree i of this power series is equal to the sum of all terms of degree i in p.  In particular, the homogeneous part of degree i of that power series is zero, whenever i is greater than the degree of p. Note that the polynomial p can be a complex constant; in that case, p is regarded as a polynomial of degree zero.

The function call PowerSeries(f, opts), where f is a procedure, creates a power series whose homogeneous part of degree i is equal to the value of the function call f(i). This assumes that the procedure f takes a non-negative integer as input and returns a polynomial as output.

If the optional argument analytic=g is passed, where g is an algebraic expression, then the power series is assumed to be equal to g in a neighbourhood of the origin. There is no verification of this assumption.

If the optional argument variables=s is passed, where s is a set of variable names, then the power series is a power series in the given variables. If this option is omitted, then the analytic option needs to be specified, and Maple determines the set of variables from that option. Either this option or the analytic option needs to be specified.

By default, every time f is invoked, its result is verified to be a polynomial of the expected homogeneous degree in the given variables. If you know this is not necessary, you can include the optional argument check=false to disable these checks for greater efficiency.

By default, every time f is invoked, its result is expanded. If you know that the value returned by f will always be expanded, you can include the optional argument expand=false to disable these calls for greater efficiency.

The function call PowerSeries(u), where u is a univariate polynomial over power series, returns u as a power series. To be precise, if v is the main variable of u and its coefficients are power series with variables in the set of variables x, let y be the union of x with v. Then the power series returned by PowerSeries(u) is the image of u in the power series ring with variables in y. This image is obtained by the natural homomorphism mapping the univariate polynomial ring R[v] into the univariate power series ring R[[v]], where R is the ring of power series in variables in x.

The function call PowerSeries(ps), where ps is a power series, returns a copy of ps.

The implementation of power series and univariate polynomials over power series relies on lazy evaluation (also known as calls-by-need), see the page MultivariatePowerSeries for details.

When using the MultivariatePowerSeries package, do not assign anything to the variables occurring in the power series and univariate polynomials over power series. If you do, you may see invalid results.

$\mathrm{with}\left(\mathrm{MultivariatePowerSeries}\right)\:$

$a≔\mathrm{PowerSeries}\left(1+x+xy+x^2\right)$

$a≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs:}1+x+x^2+xy\right]$

$b≔\mathrm{Inverse}\left(a\right)$

$b≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\frac{1}{x^2+xy+x+1}\mathrm{:}1+\mathrm{\dots }\right]$

$\mathrm{Truncate}\left(b\,5\right)$

$-2x^{4}y-3x^3{y}^2-x^4-x^{3}y+x^2{y}^2+x^3+2x^{2}y-xy-x+1$

bproc := proc(d) local s; s := 2*x + y; return expand(s^d); end proc:

$b≔\mathrm{PowerSeries}\left(\mathrm{bproc}\,\mathrm{variables}=\left\{x\,y\right\}\right)$

$b≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs:}1+\mathrm{\dots }\right]$

$\mathrm{Truncate}\left(b\,3\right)$

$8x^3+12x^{2}y+6x{y}^2+y^3+4x^2+4xy+y^2+2x+y+1$

$c≔\mathrm{PowerSeries}\left(d\mapsto \frac{x^d}{d!}\,\mathrm{analytic}=\exp \left(x\right)\right)$

$c≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}{\ⅇ}^x\mathrm{:}1+\mathrm{\dots }\right]$

$\mathrm{Truncate}\left(c\,5\right)$

$1+x+\frac{1}{2}{x}^2+\frac{1}{6}{x}^3+\frac{1}{24}{x}^4+\frac{1}{120}{x}^5$

$\mathrm{Truncate}\left(c\,10\right)$

$1+x+\frac{1}{2}{x}^2+\frac{1}{6}{x}^3+\frac{1}{24}{x}^4+\frac{1}{120}{x}^5+\frac{1}{720}{x}^6+\frac{1}{5040}{x}^7+\frac{1}{40320}{x}^8+\frac{1}{362880}{x}^9+\frac{1}{3628800}{x}^{10}$

$f≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[\mathrm{PowerSeries}\left(1\right)\,\mathrm{PowerSeries}\left(x\right)\,\mathrm{PowerSeries}\left(y\right)\,\mathrm{GeometricSeries}\left(\left[x\,y\right]\right)\right]\,z\right)\:$

$\mathrm{Truncate}\left(f\,2\right)$

$1+\left(x^2+2xy+y^2+x+y+1\right)z^3+y{z}^2+xz$

$d≔\mathrm{PowerSeries}\left(f\right)$

$d≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}1+xz+y{z}^2+\frac{z^3}{1-x-y}\mathrm{:}1+\mathrm{\dots }\right]$

$\mathrm{Truncate}\left(d\,3\right)$

$y{z}^2+z^3+xz+1$

$e≔\mathrm{PowerSeries}\left(d\right)$

$e≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}1+xz+y{z}^2+\frac{z^3}{1-x-y}\mathrm{:}1+xz+y{z}^2+z^3+\mathrm{\dots }\right]$

The MultivariatePowerSeries[PowerSeries] command was introduced in Maple 2021.

For more information on Maple 2021 changes, see Updates in Maple 2021.