15.3 Laurent Series Rings

Module: sage.rings.laurent_series_ring

Module-level Functions

LaurentSeriesRing( base_ring, [name=None])

sage: R = LaurentSeriesRing(Q, 'x'); R
Laurent Series Ring in x over Rational Field
sage: x = R.0
sage: g = 1 - x + x^2 - x^4 +O(x^8); g
1 - x + x^2 - x^4 + O(x^8)
sage: g = 10*x^(-3) + 2006 - 19*x + x^2 - x^4 +O(x^8); g
10*x^-3 + 2006 - 19*x + x^2 - x^4 + O(x^8)

You can also use more mathematical notation when the base is a field:

sage: Frac(Q[['x']])
Laurent Series Ring in x over Rational Field
sage: Frac(GF(5)['y'])
Fraction Field of Univariate Polynomial Ring in y over Finite Field of size
5

Here the fraction field is not just the Laurent series ring, so you can't use the Frac notation to make the Laurent series ring.

sage: Frac(Z[['t']])
Fraction Field of Power Series Ring in t over Integer Ring

is_LaurentSeriesRing( x)

Class: LaurentSeriesRing_domain

class LaurentSeriesRing_domain
LaurentSeriesRing_domain( self, base_ring, [name=None])

Functions: fraction_field

Class: LaurentSeriesRing_field

class LaurentSeriesRing_field
LaurentSeriesRing_field( self, base_ring, [name=None])

Class: LaurentSeriesRing_generic

class LaurentSeriesRing_generic
Univariate Laurent Series Ring

sage: K, q = LaurentSeriesRing(C, 'q').objgen(); K
Laurent Series Ring in q over Complex Field with 53 bits of precision
sage: loads(K.dumps()) == K
True
LaurentSeriesRing_generic( self, base_ring, [name=None])

Functions: base_ring,$  $ characteristic,$  $ default_prec,$  $ gen,$  $ ngens,$  $ power_series_ring,$  $ set_default_prec

base_ring( self)

sage: R = LaurentSeriesRing(QQ, "x")
sage: R.base_ring()
Rational Field
sage: S = LaurentSeriesRing(GF(17)['x'], 'y')
sage: S
Laurent Series Ring in y over Univariate Polynomial Ring in x over
Finite Field of size 17
sage: S.base_ring()
Univariate Polynomial Ring in x over Finite Field of size 17

power_series_ring( self)

If this is the Laurent series ring $ R((t))$ , return the power series ring $ R[[t]]$ .

sage: R = LaurentSeriesRing(QQ, "x")
sage: R.power_series_ring()
Power Series Ring in x over Rational Field

Special Functions: __call__,$  $ __cmp__,$  $ __reduce__,$  $ __repr__,$  $ _is_valid_homomorphism_

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