Periodic function
In
mathematics, a periodic function is a function that repeats its values after some definite
period has been added to its independent variable. Everyday examples are seen when the variable is
time; for instance the hands of a
clock or the phases of the
moon show periodic behaviour. Periodic motion is motion in which the position of the system are expressible as periodic functions, all with the
same period.
For a function on the real numbers or on the integers, that means that the entire
graph can be formed from copies of one particular portion, repeated at regular intervals.
Encyclopedia
In
mathematics, a
periodic function is a function that repeats its values after some definite
period has been added to its independent variable. Everyday examples are seen when the variable is
time; for instance the hands of a
clock or the phases of the
moon show periodic behaviour.
Periodic motion is motion in which the position of the system are expressible as periodic functions, all with the
same period.
For a function on the real numbers or on the integers, that means that the entire
graph can be formed from copies of one particular portion, repeated at regular intervals. More explicitly, a function
f is
periodic with period P greater than zero if
- f = f
for
all values of
x in the domain of
f. An
aperiodic function is one that has no such period
P.
If a function
f is periodic with period
P then for all
x in the domain of
f and all integers
n,
- f = f .
In the above example, the value of
P is 1, since
f =
f =
f = etc. The period of a function need not be the smallest value that satisfies the above equation, so
P could also equal two.
A simple example is the function
f that gives the "fractional part" of its argument:
- f = f = f = ... = 0.5.
Some named examples are
sawtooth wave,
square wave and
triangle wave.
The
trigonometric functions sine and cosine are common periodic functions, with period 2π. The subject of
Fourier series investigates the idea that an 'arbitrary' periodic function is a sum of trigonometric functions with matching periods.
A function whose domain is the
complex numbers can have two incommensurate periods without being constant. The elliptic functions are such functions.
General definition
Let
E be a set with an internal operation + . A
T-periodic function, or
function periodic with period T on
E is a function
f on
E to some set
F, such that
- for all x in E, f = f.
Note that unless + is assumed commutative this definition depends on writing
T on the right.
The period
T is not unique. For a given
T, every integer multiple of
T is also a period.
Periodic sequences
Some naturally-occurring sequences are periodic, for example the decimal expansion of any rational number . We can therefore speak of the
period or
period length of a sequence. This is just a special case of the general definition.
Translational symmetry
If a function is used to describe an object, e.g. an infinite image is given by the color as function of position, the periodicity of the function corresponds to
translational symmetry of the object.
See also
External links
