Big O for various types (power series, p-adics, etc.)¶

sage.rings.big_oh.O(*x, **kwds)

Big O constructor for various types.

EXAMPLES:

This is useful for writing power series elements:

sage: R.<t> = ZZ[['t']]
sage: (1+t)^10 + O(t^5)
1 + 10*t + 45*t^2 + 120*t^3 + 210*t^4 + O(t^5)

A power series ring is created implicitly if a polynomial element is passed:

sage: R.<x> = QQ['x']
sage: O(x^100)
O(x^100)
sage: 1/(1+x+O(x^5))
1 - x + x^2 - x^3 + x^4 + O(x^5)
sage: R.<u,v> = QQ[[]]
sage: 1 + u + v^2 + O(u, v)^5
1 + u + v^2 + O(u, v)^5

This is also useful to create $$p$$-adic numbers:

sage: O(7^6)
O(7^6)
sage: 1/3 + O(7^6)
5 + 4*7 + 4*7^2 + 4*7^3 + 4*7^4 + 4*7^5 + O(7^6)

It behaves well with respect to adding negative powers of $$p$$:

sage: a = O(11^-32); a
O(11^-32)
sage: a.parent()
11-adic Field with capped relative precision 20

There are problems if you add a rational with very negative valuation to an $$O$$-Term:

sage: 11^-12 + O(11^15)
11^-12 + O(11^8)

The reason that this fails is that the constructor doesn’t know the right precision cap to use. If you cast explicitly or use other means of element creation, you can get around this issue:

sage: K = Qp(11, 30)
sage: K(11^-12) + O(11^15)
11^-12 + O(11^15)
sage: 11^-12 + K(O(11^15))
11^-12 + O(11^15)
sage: K(11^-12, absprec = 15)
11^-12 + O(11^15)
sage: K(11^-12, 15)
11^-12 + O(11^15)

We can also work with asymptotic expansions:

sage: A.<n> = AsymptoticRing(growth_group='QQ^n * n^QQ * log(n)^QQ', coefficient_ring=QQ); A
Asymptotic Ring <QQ^n * n^QQ * log(n)^QQ * Signs^n> over Rational Field
sage: O(n)
O(n)