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Mathematics LibreTexts

4.3: Periodic Functions

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    We first need to define a periodic function. A function is called periodic with period \(p\) if \(f(x+p)=f(x)\), for all \(x\), even if \(f\) is not defined everywhere. A simple example is the function f(x)=sin(bx) which is periodic with period (2π)∕b. Of course it is also periodic with periodic (4π)∕b. In general a function with period p is periodic with period 2p3p…. This can easily be seen using the definition of periodicity, which subtracts p from the argument

    \[ f(x+3p) = f(x+2p) = f(x+p) = f(x).\]

    The smallest positive value of p for which f is periodic is called the (primitive) period of f.

    Exercise \(\PageIndex{1}\)

    Question: What is the primitive period of \(\sin(4x)\)?