6.15.2.1 LaurentSeries Objects

class LaurentSeries
LaurentSeries( parent, f, [n=0])

Create the Laurent series $ t^n \cdot f$ . The default is n=0.

INPUT:
    parent -- a Laurent series ring
    f -- a power series (or something can be coerced to one)
    n -- integer (default 0)

OUTPUT:
    a Laurent series

...

Instances of class LaurentSeries have the following methods (in addition to inherited methods and special methods):

add_bigoh,$  $ copy,$  $ degree,$  $ derivative,$  $ integral,$  $ is_zero,$  $ power_series,$  $ prec,$  $ unit_part,$  $ valuation,$  $ variable

Further documentation:

derivative( )

The derivative of this Laurent series.

integral( )

The integral of this Laurent series with 0 constant term.

The integral may or may not be defined if the base ring is not a field.

sage: t = LaurentSeriesRing(IntegerRing(), 't').gen()
sage: f = 2*t**-3 + 3*t**2 + O(t**4)
sage: f.integral()
-t^-2 + t^3 + O(t^5)

sage: f = t**3
sage: f.integral()
Traceback (most recent call last):
...
ArithmeticError: Coefficients of integral of t^3 cannot be coerced into the
base ring

The integral of 1/t is $ \log(t)$ , which is not given by a Laurent series:

sage: t = LaurentSeriesRing(RationalField(), 't').gen()
sage: f = -1/t**3 - 31/t + O(t**3)
sage: f.integral()
Traceback (most recent call last):
...
ArithmeticError: The integral of -t^-3 - 31*t^-1 + O(t^3) is not a Laurent
series, since t^-1 has nonzero coefficient -31.

prec( )

This function returns the n so that the Laurent series is of the form (stuff) + $ O(t^n)$ . It doesn't matter how many negative powers appear in the expansion. In particular, prec could be negative.

Instances of class LaurentSeries also have the following special methods:

__add__,$  $ __call__,$  $ __cmp__,$  $ __div__,$  $ __getitem__,$  $ __mul__,$  $ __neg__,$  $ __pow__,$  $ __repr__,$  $ __setitem__,$  $ __sub__,$  $ _latex_,$  $ _LaurentSeries__normalize

Further documentation:

_LaurentSeries__normalize( )

A Laurent series is a pair (u(t), n), where either u=0 (to some precision) or u is a unit. This pair corresponds to $ t^n\cdot u(t)$ .

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