Everything about Oscillation Mathematics totally explained
In
mathematics,
oscillation is the behaviour of a
sequence of
real numbers or a real-valued
function, which doesn't
converge, but also doesn't
diverge to +∞ or -∞; that is, oscillation is the failure to have a
limit, and is also a quantitative measure for that.
Oscillation is defined as the difference (possibly ∞) between the
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's zero if and only if the sequence converges. For a function, the oscillation is defined at every
limit point in [-∞,+∞] of the
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 at that point.
Examples
- 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
periodic, and any sequence that's periodic without being constant will have non-zero oscillation. On the other hand, non-zero oscillation doesn't 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 f is defined at each
x ∈
X by
» Further Information
Get more info on 'Oscillation Mathematics'.
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