4.11.2.1 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):

derivative,$  $ integral,$  $ prec

These methods are defined as follows:

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()).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()).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.

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