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# 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 $$\PageIndex{1}$$

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

Answer

$$\frac{π}{2}$$.