Global: BoolElt Default: true
Precision: RngIntElt Default: 20
Given a commutative ring R, create the ring R[[x]] of formal power series over R. If a second integer argument p is given, the resulting ring is a fixed precision series ring with fixed precision p; otherwise the resulting ring is a free precision series ring and the optional argument Precision may be used to set the default precision for elements of the power series ring (it will be 20 otherwise; see the section above on free and fixed precision). By default, a global series ring will be returned; if the parameter Global is set to false, a non-global series ring will be returned (to which a separate name for the indeterminate can be assigned). The angle bracket notation can be used to assign a name to the indeterminate: S<x> := PowerSeriesRing(R).
Global: BoolElt Default: true
Precision: RngIntElt Default: 20
Given a commutative ring R, create the ring R((x)) of formal Laurent series over R. If a second integer argument p is given, the resulting ring is a fixed precision series ring with fixed precision p; otherwise the resulting ring is a free precision series ring and the optional argument Precision may be used to set the default precision for elements of the power series ring (it will be 20 otherwise; see the section above on free and fixed precision). By default, a global series ring will be returned; if the parameter Global is set to false, a non-global series ring will be returned (to which a separate name for the indeterminate can be assigned). The angle bracket notation can be used to assign a name to the indeterminate: S<x> := LaurentSeriesRing(R).
Global: BoolElt Default: true
Precision: RngIntElt Default: 20
Given a commutative ring R, create the ring R< < x > > of formal Puiseux series over R. If a second integer argument p is given, the resulting ring is a fixed precision series ring with fixed precision p; otherwise the resulting ring is a free precision series ring and the optional argument Precision may be used to set the default precision for elements of the power series ring (it will be 20 otherwise; see the section above on free and fixed precision). The optional argument Precision may be used to set the default precision for elements of the power series ring. By default, a global series ring will be returned; if the parameter Global is set to false, a non-global series ring will be returned (to which a separate name for the indeterminate can be assigned). The angle bracket notation can be used to assign a name to the indeterminate: S<x> := PuiseuxSeriesRing(R).
> Q := RationalField(); > P<x> := PowerSeriesRing(Q); > PP := PowerSeriesRing(Q); > P; Power series ring in x over Rational Field > PP; Power series ring in x over Rational Field > PP.1; xPP is identical to P. We now create non-global series rings (which are also different to the global series ring). Note that elements of all the rings are mathematically equal by automatic coercion.
> Pa<a> := PowerSeriesRing(Q: Global := false); > Pb<b> := PowerSeriesRing(Q: Global := false); > Pa; Power series ring in a over Rational Field > Pb; Power series ring in b over Rational Field > a; a > b; b > P; Power series ring in x over Rational Field > x; x > x eq a; // Automatic coercion true > x + a; 2*x
Procedure to change the default precision on a free series ring series S; the default precision will be set to n, which must be a non-negative integer.
Function that returns a Boolean indicating whether a default precision has been set on the free series ring S (which will always be true), as well as its (non-negative) integer value (which is 20 by default).
Procedure to change the name of the `indeterminate' transcendental element generating the series ring or field S; the name (used in printing elements of S) is changed to the string x. Note that no assignment to the identifier x is made (so x cannot be used for the specification of elements of S without further assignment).
Return the element of the series ring or field with a name attached to it, that is, return the `indeterminate' transcendental element generating S over its coefficient ring.
The easiest way to create power and Laurent series in a given ring is to use the angle bracket construction to attach names to the indeterminate, and to use these names to express the series (see the examples). Below we list other options.
Return the generator (indeterminate) for the series ring R.
Given a series ring R, integers v and p (where p > 0 or p = ∞), and a sequence a=[a1, ..., ad] of elements of R, create the element in R with valuation v, known coefficients given by a and relative precision p. That is, this function returns the series a1xv + ... + adxv + d - 1 + O(xv + p), or, if p = (-1), the exact series a1xv + ... + adxv + d - 1. If R is a power series ring, then v must be non-negative.The integer v or the integer p or both may be omitted. If v is omitted, it will be set to zero by default; if p is omitted it will be taken to be v + d, where d is the length of the sequence a.
Coerce s into the series ring R. Here s is allowed to be a sequence of elements from (or coercible into) the coefficient ring of R, or just an element from (or coercible into) R. A sequence [a1, ..., ad] is converted into the series a1 + a2x1 + ... + adxd - 1 + O(xd).
Create the series O(xv) where x is the generator of the parent of f and v is the valuation of f. The most typical usage of this function is the expression O(xn) where x is the generator of a series ring, but a general series f is actually allowed.