EXAMPLES:
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
...
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
Saving and loading.
sage: loads(q.dumps()) == q
True
sage: loads(K.dumps()) == K
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.
AUTHORS:
Bases: sage.structure.element.AlgebraElement
A Laurent Series.
EXAMPLES:
sage: R.<t> = LaurentSeriesRing(QQ)
sage: f = t^2 + t^3 + O(t^10); f
t^2 + t^3 + O(t^10)
sage: f.add_bigoh(5)
t^2 + t^3 + O(t^5)
Return the nonzero coefficients of self.
EXAMPLES:
sage: R.<t> = LaurentSeriesRing(QQ)
sage: f = -5/t^(2) + t + t^2 - 10/3*t^3
sage: f.coefficients()
[-5, 1, 1, -10/3]
Returns minimum precision of and self.
EXAMPLES:
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
Return the degree of a polynomial equivalent to this power series modulo big oh of the precision.
EXAMPLES:
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
The formal derivative of this Laurent series, with respect to variables supplied in args.
Multiple variables and iteration counts may be supplied; see documentation for the global derivative() function for more details.
See also
_derivative()
EXAMPLES:
sage: R.<x> = LaurentSeriesRing(QQ)
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)
sage: g.derivative(x)
-10*x^-11 - 1 + 2*x - 4*x^3 + O(x^7)
sage: R.<t> = PolynomialRing(ZZ)
sage: S.<x> = LaurentSeriesRing(R)
sage: f = 2*t/x + (3*t^2 + 6*t)*x + O(x^2)
sage: f.derivative()
-2*t*x^-2 + (3*t^2 + 6*t) + O(x)
sage: f.derivative(x)
-2*t*x^-2 + (3*t^2 + 6*t) + O(x)
sage: f.derivative(t)
2*x^-1 + (6*t + 6)*x + O(x^2)
Return the exponents appearing in self with nonzero coefficients.
EXAMPLES:
sage: R.<t> = LaurentSeriesRing(QQ)
sage: f = -5/t^(2) + t + t^2 - 10/3*t^3
sage: f.exponents()
[-2, 1, 2, 3]
The formal integral of this Laurent series with 0 constant term.
EXAMPLES: 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()
...
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()
...
ArithmeticError: The integral of is not a Laurent series, since t^-1 has nonzero coefficient.
Another example with just one negative coefficient:
sage: A.<t> = QQ[[]]
sage: f = -2*t^(-4) + O(t^8)
sage: f.integral()
2/3*t^-3 + O(t^9)
sage: f.integral().derivative() == f
True
Returns True if this is Laurent series is a unit in this ring.
EXAMPLES:
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
...
ArithmeticError: division not defined
ALGORITHM: A Laurent series is a unit if and only if its “unit part” is a unit.
EXAMPLES:
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
EXAMPLES:
sage: R.<t> = LaurentSeriesRing(QQ)
sage: f = -5/t^(2) + t + t^2 - 10/3*t^3
sage: f.list()
[-5, 0, 0, 1, 1, -10/3]
EXAMPLES:
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()
...
ArithmeticError: self is a not a power series
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.
EXAMPLES:
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
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 consistent with polynomials, integers, etc.
EXAMPLES:
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)
AUTHORS:
Returns the laurent series equivalent to self except without any degree n terms.
This is equivalent to `self - self.truncate(n)`.
EXAMPLES:
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
EXAMPLES:
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)
EXAMPLES:
sage: x = Frac(QQ[['x']]).0
sage: f = 1/x + x^2 + 3*x^4 + O(x^7)
sage: f.variable()
'x'