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28.3 Laurent Series Rings

Module: sage.rings.laurent_series_ring

Laurent Series Rings 

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

Module-level Functions

LaurentSeriesRing( base_ring, [name=None], [names=None], [sparse=False])

sage: R = LaurentSeriesRing(QQ, '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(QQ[['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(ZZ[['t']])
Fraction Field of Power Series Ring in t over Integer Ring

Laurent series rings are determined by their variable and the base ring, and are globally unique. 

sage: K = Qp(5, prec = 5)
sage: L = Qp(5, prec = 200)
sage: R.<x> = LaurentSeriesRing(K)
sage: S.<y> = LaurentSeriesRing(L)
sage: R is S
False
sage: T.<y> = LaurentSeriesRing(Qp(5,prec=200))
sage: S is T
True
sage: W.<y> = LaurentSeriesRing(Qp(5,prec=199))
sage: W is T
False

is_LaurentSeriesRing( x)

Class: LaurentSeriesRing_domain

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

Class: LaurentSeriesRing_field

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

Class: LaurentSeriesRing_generic

class LaurentSeriesRing_generic
Univariate Laurent Series Ring 

sage: K, q = LaurentSeriesRing(CC, '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], [sparse=False])

Functions: base_extend,$ $ change_ring,$ $ characteristic,$ $ default_prec,$ $ gen,$ $ is_dense,$ $ is_exact,$ $ is_field,$ $ is_sparse,$ $ ngens,$ $ power_series_ring,$ $ set_default_prec

base_extend( self, R)
Returns the laurent series ring over R in the same variable as self, assuming there is a canonical coerce map from the base ring of self to R.

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__,$ $ _coerce_impl,$ $ _is_valid_homomorphism_

__call__( self, x, [n=0])

sage: R.<u> = LaurentSeriesRing(Qp(5, 10))
sage: S.<t> = LaurentSeriesRing(RationalField())
sage: print R(t + t^2 + O(t^3))
(1 + O(5^10))*u + (1 + O(5^10))*u^2 + O(u^3)

Note that coercing an element into its own parent just produces that element again (since Laurent series are immutable): 

sage: u is R(u)
True

_coerce_impl( self, x)
Return canonical coercion of x into self.

Rings that canonically coerce to this power series ring R:

* R itself * Any laurent series ring in the same variable whose base ring canonically coerces to the base ring of R. * Any ring that canonically coerces to the power series ring over the base ring of R. * Any ring that canonically coerces to the base ring of R

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Release 2007.10.28, documentation updated on October 28, 2007.
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