for all values of x in the domain of f.
A simple example is the function f that gives the "fractional part" of its argument:
If a function f is periodic with period t the for all x in the domain of f and all integers n,
In the above example, the value of t is 1, since f( x ) = f( x + 1 ) = f( x + 2 ) ...
Some named examples are sawtooth wave, triangle wave.
Sine and cosine are periodic functions, with period 2π. The subject of Fourier series investigates the idea that an 'arbitrary' periodic function is a sum of trigometric 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. ("Incommensurate" in this context means not real multiples of each other.)
Let E be a set with a + internal operation. Let f be a function from E to F.
f is said Tperiodic (or periodic with period T) iff ∃ T in E such that ∀ x in E, f(x+T) = f(x)
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