order of vanishing

Definition.  Let x0 be a zero (http://planetmath.org/ZeroOfAFunction) of the real function Δ.  The order of vanishing of Δ at x0 is n, if  limxx0Δ(x)xn has a non-zero finite value.

Usually, x0 of the definition is 0.

Example.  If the curves   y=f(x)  and  y=g(x)  have in the point  (x0,y0)  the order of contact n, then the differencePlanetmathPlanetmathΔ(h):=g(x0+h)-f(x0+h)  of the ordinates has n+1-order of vanishing.

Title order of vanishing
Canonical name OrderOfVanishing
Date of creation 2013-03-22 17:57:15
Last modified on 2013-03-22 17:57:15
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 6
Author pahio (2872)
Entry type Definition
Classification msc 26E99
Synonym vanishing order
Related topic Multiplicity
Related topic OsculatingCurve