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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(s) of the system are expressible as periodic functions, all with the same period. Euclid, detail from The School of Athens by Raphael. ...
Partial plot of a function f. ...
A wall clock A clock (from the Latin cloca, bell) is an instrument for measuring time. ...
Bulk composition of the moons mantle and crust estimated, weight percent Oxygen 42. ...
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 In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...
- f(x + P) = f(x)
for all values of x in the domain of f. An aperiodic function (non-periodic function) is one that has no such period P.
A simple example is the function f that gives the "fractional part" of its argument:
- f( 0.5 ) = f( 1.5 ) = f( 2.5 ) = ... = 0.5.
If a function f is periodic with period P then for all x in the domain of f and all integers n, - f( x + Pn ) = f ( x ).
In the above example, the value of P is 1, since f( x ) = f( x + 1 ) = f( x + 2 ) = etc. The period of a function need not be the smallest value (least period) that satisfies the above equation, so P could also equal two.
Some named examples are sawtooth wave, square wave and triangle wave. The sawtooth wave (or saw wave) is a kind of basic non-sinusoidal waveform. ...
A square wave is a kind of basic waveform. ...
A triangle wave is a waveform named for its triangular shape. ...
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. In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...
Fourier series are a mathematical technique for analyzing an arbitrary periodic function by decomposing the function into a sum of much simpler sinusoidal component functions, which differ from each other only in amplitude and frequency. ...
A function whose domain is the complex numbers can have two incommensurate periods without being constant. The elliptic functions are such functions. ("Incommensurate" in this context means not real multiples of each other.) Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ...
In complex analysis, an elliptic function is, roughly speaking, a function defined on the complex plane which is periodic in two directions. ...
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 The word operation can mean any of several things: The method, act, process, or effect of using a device or system. ...
Partial plot of a function f. ...
- for all x in E, f(x + T) = f(x).
Note that unless + is assumed commutative this definition depends on writing T on the right. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...
The period T is not unique. For a given T, every multiple of T is also a period. If there does not exist a smallest period T0, then the function is constant. In mathematics a constant function is a function whose values do not vary and thus are constant. ...
Periodic sequences Some naturally-occurring sequences are periodic, for example (eventually) the decimal expansion of any rational number (see recurring decimal). We can therefore speak of the period or period length of a sequence. This is (if one insists) just a special case of the general definition. In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...
The decimal (base ten or occasionally denary) numeral system has ten as its base. ...
In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...
A recurring decimal is an expression representing a real number in the decimal numeral system, in which after some point the same sequence of digits repeats infinitely many times. ...
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. Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ...
See also In mathematics, almost periodicity is a property of dynamical systems that appear to retrace their paths through phase space, but not exactly. ...
Amplitude is a nonnegative scalar measure of a waves magnitude of oscillation, that is, magnitude of the maximum disturbance in the medium during one wave cycle. ...
In music a sound or note of definite pitch is one of which it is possible or relatively easy to discern the pitch or frequency of the fundamental, as opposed to sounds of indefinite pitch. ...
Sine waves of various frequencies; the lower waves have higher frequencies than those above. ...
Oscillation is the periodic variation, typically in time, of some measure as seen, for example, in a swinging pendulum. ...
The wavelength is the distance between repeating units of a wave pattern. ...
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