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[parent] order of vanishing (Definition)

Definition. Let $ x_0$ be a zero of the real function $ \Delta$. The order of vanishing of $ \Delta$ at $ x_0$ is $ n$, if $ \displaystyle\lim_{x\to x_0}\frac{\Delta(x)}{x^n}$ has a non-zero finite value.

Usually, $ x_0$ of the definition is 0.

Example. If the curves $ y = f(x)$ and $ y = g(x)$ have in the point $ (x_0,\,y_0)$ the order of contact $ n$, then the difference $ \Delta(h) := g(x_0+h)-f(x_0+h)$ of the ordinates has $ n\!+\!1$-order of vanishing.



"order of vanishing" is owned by pahio.
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See Also: multiplicity, osculating curve

Other names:  vanishing order

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Cross-references: ordinates, difference, order of contact, point, curves, finite, real function

This is version 3 of order of vanishing, born on 2008-03-28, modified 2008-03-29.
Object id is 10453, canonical name is OrderOfVanishing.
Accessed 346 times total.

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AMS MSC: 26E99 (Real functions :: Miscellaneous topics :: Miscellaneous)
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