properties of O and o

The following properties of Landau notationPlanetmathPlanetmath hold:

  1. 1.

    o(f) and O(f) are vector spaces, i.e. if g,ho(f) (resp. in O(f)) then λg+μho(f) (resp. in O(f)) whenever λ,μ; In particular o(f)+o(f)=o(f) and λo(f)=o(f);

  2. 2.

    if λ0 then λo(f)=o(f) and λO(f)=O(f);

  3. 3.

    fo(g)=o(fg), fO(g)=O(fg);

  4. 4.

    o(g)α=o(gα), O(g)α=O(gα);

  5. 5.

    o(f)O(f); on the other hand if fo(g) then O(f)o(g);

  6. 6.

    o(f)o(g) if fO(g); analogously O(f)O(g) if fO(g);

  7. 7.

    o(o(f))=o(f), O(O(f))=O(f), o(O(f))=o(f), O(o(f))=o(f).

Here are some examples. First of all we consider Taylor formula. If x0(a,b) and f:(a,b) has n derivatives, then


As a consequence, if f has n+1 derivatives, we can replace o((x-x0)n) with O((x-x0)n+1) in the previous formula.

For example:


Using the properties stated above we can compose and iterate Taylor expansionsMathworldPlanetmath. For example from the expansions


we get

(xsinx-e(x2))log(cosx) (x(x-x33!+o(x4))-(1+x2+x42+O((x2)3))log(1-x22+x44!+o(x5))
Title properties of O and o
Canonical name PropertiesOfOAndO
Date of creation 2013-03-22 15:15:45
Last modified on 2013-03-22 15:15:45
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 7
Author paolini (1187)
Entry type Result
Classification msc 26A12
Related topic FormalDefinitionOfLandauNotation