# 28.4 Laurent Series

Module: sage.rings.laurent_series_ring_element

File: sage/rings/laurent_series_ring_element.pyx (starting at line 1)

Laurent Series

sage: R.<t> = LaurentSeriesRing(GF(7), 't'); R
Laurent Series Ring in t over Finite Field of size 7
sage: f = 1/(1-t+O(t^10)); f
1 + t + t^2 + t^3 + t^4 + t^5 + t^6 + t^7 + t^8 + t^9 + O(t^10)


Laurent series are immutable:

sage: f[2]
1
sage: f[2] = 5
Traceback (most recent call last):
...
IndexError: Laurent series are immutable


We compute with a Laurent series over the complex mpfr numbers.

sage: K.<q> = Frac(CC[['q']])
sage: K
Laurent Series Ring in q over Complex Field with 53 bits of precision
sage: q
1.00000000000000*q


sage: loads(q.dumps()) == q
True
True


IMPLEMENTATION: Laurent series in SAGE are represented internally as a power of the variable times the unit part (which need not be a unit - it's a polynomial with nonzero constant term). The zero Laurent series has unit part 0.

Author Log:

• William Stein: original version
• David Joyner: added examples 2006-01-22
• Robert Bradshaw: optimizations, shifting 2007-04

Module-level Functions

 is_LaurentSeries( )

 make_element_from_parent( )

Class: LaurentSeries

class LaurentSeries
File: sage/rings/laurent_series_ring_element.pyx (starting at line 65)

A Laurent Series.

Functions: add_bigoh, change_ring, common_prec, copy, degree, derivative, integral, is_unit, is_zero, list, power_series, prec, shift, truncate, truncate_neg, valuation, valuation_zero_part, variable

sage: R.<t> = LaurentSeriesRing(QQ)
sage: f = t^2 + t^3 + O(t^10); f
t^2 + t^3 + O(t^10)
t^2 + t^3 + O(t^5)


 common_prec( )

Returns minimum precision of and self.

sage: R.<t> = LaurentSeriesRing(QQ)


sage: f = t^(-1) + t + t^2 + O(t^3)
sage: g = t + t^3 + t^4 + O(t^4)
sage: f.common_prec(g)
3
sage: g.common_prec(f)
3


sage: f = t + t^2 + O(t^3)
sage: g = t^(-3) + t^2
sage: f.common_prec(g)
3
sage: g.common_prec(f)
3


sage: f = t + t^2
sage: f = t^2
sage: f.common_prec(g)
+Infinity


sage: f = t^(-3) + O(t^(-2))
sage: g = t^(-5) + O(t^(-1))
sage: f.common_prec(g)
-2


 degree( )

Return the degree of a polynomial equivalent to this power series modulo big oh of the precision.

sage: x = Frac(QQ[['x']]).0
sage: g = x^2 - x^4 + O(x^8)
sage: g.degree()
4
sage: g = -10/x^5 + x^2 - x^4 + O(x^8)
sage: g.degree()
4


 derivative( )

The formal derivative of this Laurent series.

sage: x = Frac(QQ[['x']]).0
sage: f = x^2 + 3*x^4 + O(x^7)
sage: f.derivative()
2*x + 12*x^3 + O(x^6)
sage: g = 1/x^10 - x + x^2 - x^4 + O(x^8)
sage: g.derivative()
-10*x^-11 - 1 + 2*x - 4*x^3 + O(x^7)


 integral( )

The formal 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(ZZ, 't').0
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 cannot be coerced into the base
ring


The integral of 1/t is , which is not given by a Laurent series:

sage: t = Frac(QQ[['t']]).0
sage: f = -1/t^3 - 31/t + O(t^3)
sage: f.integral()
Traceback (most recent call last):
...
ArithmeticError: The integral of is not a Laurent series, since t^-1 has
nonzero coefficient.


 is_unit( )

Returns True if this is Laurent series is a unit in this ring.

sage: R.<t> = LaurentSeriesRing(QQ)
sage: (2+t).is_unit()
True
sage: f = 2+t^2+O(t^10); f.is_unit()
True
sage: 1/f
1/2 - 1/4*t^2 + 1/8*t^4 - 1/16*t^6 + 1/32*t^8 + O(t^10)
sage: R(0).is_unit()
False
sage: R.<s> = LaurentSeriesRing(ZZ)
sage: f = 2 + s^2 + O(s^10)
sage: f.is_unit()
False
sage: 1/f
Traceback (most recent call last):
...
ArithmeticError: division not defined


ALGORITHM: A Laurent series is a unit if and only if its "unit part" is a unit.

 is_zero( )

sage: x = Frac(QQ[['x']]).0
sage: f = 1/x + x + x^2 + 3*x^4 + O(x^7)
sage: f.is_zero()
0
sage: z = 0*f
sage: z.is_zero()
1


 power_series( )

sage: R.<t> = LaurentSeriesRing(ZZ)
sage: f = 1/(1-t+O(t^10)); f.parent()
Laurent Series Ring in t over Integer Ring
sage: g = f.power_series(); g
1 + t + t^2 + t^3 + t^4 + t^5 + t^6 + t^7 + t^8 + t^9 + O(t^10)
sage: parent(g)
Power Series Ring in t over Integer Ring
sage: f = 3/t^2 +  t^2 + t^3 + O(t^10)
sage: f.power_series()
Traceback (most recent call last):
...
ArithmeticError: self is a not a power series


 prec( )

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

sage: x = Frac(QQ[['x']]).0
sage: f = x^2 + 3*x^4 + O(x^7)
sage: f.prec()
7
sage: g = 1/x^10 - x + x^2 - x^4 + O(x^8)
sage: g.prec()
8


 shift( )

Returns this laurent series multiplied by the power . Does not change this series.

NOTE: Despite the fact that higher order terms are printed to the right in a power series, right shifting decreases the powers of , while left shifting increases them. This is to be consistant with polynomials, integers, etc.

sage: R.<t> = LaurentSeriesRing(QQ['y'])
sage: f = (t+t^-1)^4; f
t^-4 + 4*t^-2 + 6 + 4*t^2 + t^4
sage: f.shift(10)
t^6 + 4*t^8 + 6*t^10 + 4*t^12 + t^14
sage: f >> 10
t^-14 + 4*t^-12 + 6*t^-10 + 4*t^-8 + t^-6
sage: t << 4
t^5
sage: t + O(t^3) >> 4
t^-3 + O(t^-1)


 truncate( )

Returns the laurent series of degree which is equivalent to self modulo .

 truncate_neg( )

Returns the laurent series equivalent to self except without any degree < n terms.

This is equivalent to .

 valuation( )

sage: x = Frac(QQ[['x']]).0
sage: f = 1/x + x^2 + 3*x^4 + O(x^7)
sage: g = 1 - x + x^2 - x^4 + O(x^8)
sage: f.valuation()
-1
sage: g.valuation()
0


 valuation_zero_part( )

sage: x = Frac(QQ[['x']]).0
sage: f = x + x^2 + 3*x^4 + O(x^7)
sage: f/x
1 + x + 3*x^3 + O(x^6)
sage: f.valuation_zero_part()
1 + x + 3*x^3 + O(x^6)
sage: g = 1/x^7 - x + x^2 - x^4 + O(x^8)
sage: g.valuation_zero_part()
1 - x^8 + x^9 - x^11 + O(x^15)


 variable( )

sage: x = Frac(QQ[['x']]).0
sage: f = 1/x + x^2 + 3*x^4 + O(x^7)
sage: f.variable()
'x'


Special Functions: __call__, __delitem__, __eq__, __ge__, __getitem__, __getslice__, __gt__, __iter__, __le__, __lshift__, __lt__, __ne__, __neg__, __normalize, __pow__, __reduce__, __rlshift__, __rpow__, __rrshift__, __rshift__, __setitem__, _im_gens_, _latex_, _repr_, _unsafe_mutate

 __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 .

 __reduce__( )

 _im_gens_( )

 _latex_( )

sage: x = Frac(QQ[['x']]).0
sage: f = (17/2)*x^-2 + x + x^2 + 3*x^4 + O(x^7)
sage: latex(f)
\frac{\frac{17}{2}}{x^{2}} + x + x^{2} + 3x^{4} + O(\text{x}^{7})


 _repr_( )

sage: R.<t> = LaurentSeriesRing(QQ)
sage: (2 + (2/3)*t^3).__repr__()
'2 + 2/3*t^3'


 _unsafe_mutate( )

SAGE assumes throughout that commutative ring elements are immutable. This is relevant for caching, etc. But sometimes you need to change a Laurent series and you really know what you're doing. That's when this function is for you.