Module: sage.rings.laurent_series_ring_element
Author Log:
Class: LaurentSeries
self, parent, f, [n=0]) |
Create the Laurent series
. 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
sage: K, q = Frac(C[['q']]).objgen() sage: K Laurent Series Ring in q over Complex Field with 53 bits of precision sage: q 1.0000000000000000*q
Saving and loading.
sage: loads(q.dumps()) == q True sage: loads(K.dumps()) == K True
Functions: add_bigoh,
copy,
degree,
derivative,
integral,
is_zero,
power_series,
prec,
unit_part,
valuation,
variable
self) |
Return the degree of a polynomial equivalent to this power series modulo big oh of the precision.
sage: x = Frac(Q[['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
self) |
The formal derivative of this Laurent series.
sage: x = Frac(Q[['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)
self) |
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(Z, '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 of t^3 cannot be coerced into the base ring
The integral of 1/t is
, which is not given by a Laurent series:
sage: t = Frac(Q[['t']]).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.
self) |
sage: x = Frac(Q[['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
self) |
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(Q[['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
self) |
sage: x = Frac(Q[['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.unit_part() 1 + x + 3*x^3 + O(x^6) sage: g = 1/x^7 - x + x^2 - x^4 + O(x^8) sage: g.unit_part() 1 - x^8 + x^9 - x^11 + O(x^15)
self) |
sage: x = Frac(Q[['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
self) |
sage: x = Frac(Q[['x']]).0 sage: f = 1/x + x^2 + 3*x^4 + O(x^7) sage: f.variable() 'x'
Special Functions: __add__,
__call__,
__div__,
__getitem__,
__getslice__,
__iter__,
__mul__,
__neg__,
__pow__,
__repr__,
__setitem__,
__sub__,
_cmp_,
_im_gens_,
_latex_,
_LaurentSeries__normalize
self, right) |
sage: x = Frac(Q[['x']]).0 sage: f = 1/x^10 + x + x^2 + 3*x^4 + O(x^7) sage: g = 1 - x + x^2 - x^4 + O(x^8) sage: f*g x^-10 - x^-9 + x^-8 - x^-6 + O(x^-2)
self, x) |
Compute value of this Laurent series at x.
sage: t = LaurentSeriesRing(Z, 't').0 sage: f = t^(-2) + t^2 + O(t^8) sage: f(2) 17/4 sage: f(-1) 2 sage: f(1/3) 82/9
self, right) |
sage: x = Frac(Q[['x']]).0 sage: f = x + x^2 + 3*x^4 + O(x^7) sage: g = 1/x^7 - x + x^2 - x^4 + O(x^8) sage: f/x 1 + x + 3*x^3 + O(x^6) sage: f/g x^8 + x^9 + 3*x^11 + O(x^14)
self, right) |
sage: x = Frac(Q[['x']]).0 sage: f = 1/x^3 + x + x^2 + 3*x^4 + O(x^7) sage: g = 1 - x + x^2 - x^4 + O(x^8) sage: f*g x^-3 - x^-2 + x^-1 + 4*x^4 + O(x^5)
self, right) |
sage: x = Frac(Q[['x']]).0 sage: f = x + x^2 + 3*x^4 + O(x^7) sage: g = 1/x^10 - x + x^2 - x^4 + O(x^8) sage: f^7 x^7 + 7*x^8 + 21*x^9 + 56*x^10 + 161*x^11 + 336*x^12 + O(x^13) sage: g^7 x^-70 - 7*x^-59 + 7*x^-58 - 7*x^-56 + O(x^-52)
self, right) |
sage: x = Frac(Q[['x']]).0 sage: f = x + x^2 + 3*x^4 + O(x^7) sage: g = 1/x^7 - x + x^2 - x^4 + O(x^8) sage: f<g False sage: f>g True
self) |
sage: x = Frac(Q[['x']]).0 sage: f = (17/2)*x^-2 + x + x^2 + 3*x^4 + O(x^7) sage: f._latex_() '\frac{17}{2}x^{-2} + x + x^{2} + 3x^{4} + \cdots'
self) |
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
.
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