Oscillation (mathematics)
475449
213580900
2008-05-19T23:14:15Z
Pbroks13
2308181
SVG image
[[Image:LimSup.svg|right|thumb|300px|Oscillation of a sequence (shown in blue) is the difference between the limit superior and limit inferior of the sequence.]]
In [[mathematics]], '''oscillation''' is the behaviour of a [[sequence]] of [[real number]]s or a real-valued [[function (mathematics)|function]], which does not [[convergence|converge]], but also does not [[divergent series|diverge]] to +∞ or -∞; that is, oscillation is the failure to have a [[Limit (mathematics)|limit]], and is also a quantitative measure for that.
Oscillation is defined as the difference (possibly ∞) between the [[Limit superior and limit inferior|limit superior and limit inferior]]. It is undefined if both are +∞ or both are -∞ (that is, if the sequence or function tends to +∞ or -∞). For a sequence, the oscillation is defined at infinity, it is zero if and only if the sequence converges. For a function, the oscillation is defined at every [[limit point]] in [-∞, +∞] of the [[domain (mathematics)|domain]] of the function (apart from the mentioned restriction). It is zero at a point if and only if the function has a finite [[Limit of a function|limit]] at that point.
==Examples==
[[Image:Rapid Oscillation.svg|thumb|300px|right|As ''ƒ''(x) approaches point ''P'', it oscillates from ''ƒ''(a) to ''ƒ''(b) infinitely many times, and does not converge.]]
*1/''x'' has oscillation ∞ at ''x'' = 0, and oscillation 0 at other finite ''x'' and at -∞ and +∞.
*sin (1/''x'') has oscillation 2 at ''x'' = 0, and 0 elsewhere.
*sin ''x'' has oscillation 0 at every finite ''x'', and 2 at -∞ and +∞.
*The sequence 1, −1, 1, −1, 1, −1, ... has oscillation 2.
In the last example the sequence is [[periodicity|periodic]], and any sequence that is periodic without being constant will have non-zero oscillation. On the other hand, non-zero oscillation does not imply periodicity.
Geometrically, the graph of an oscillating function on the real numbers follows some path in the ''xy''-plane, without settling into ever-smaller regions. In [[well-behaved]] cases the path might look like a loop coming back on itself, that is, periodic behaviour; in the worst cases quite irregular movement covering a whole region.
==Generalizations==
More generally, if ''f'' : ''X'' → ''Y'' is a function from a [[topological space]] ''X'' into a [[metric space]] ''Y'', then the '''oscillation of <i>f</i>''' is defined at each ''x'' ∈ ''X'' by
:<math>\omega(x) = \inf\left\{\mathrm{diam}(f(U))\mid U\mathrm{\ is\ a\ neighborhood\ of\ }x\right\}</math>
== See also ==
* [[Grandi's series]]
* [[Bounded mean oscillation]]
==References==
*{{cite book|author=Hewitt and Stromberg|title=Real and abstract analysis|page=78|publisher=Springer-Verlag|year=1965}}
*{{cite book|author=Oxtoby, J|title=Measure and category|publisher=Springer-Verlag|edition=4th ed.|year=1996|pages=31-35|isbn=978-0387905082}}
*{{cite book
| last = Pugh
| first = C. C.
| title = Real mathematical analysis
| publisher = New York: Springer
| date = 2002
| pages = pages 164 — 165
| isbn = 0387952977
}}
[[Category:Real analysis]]