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Definition
Let $L > 0 $ be a real number. A function $f: \R \to \C $ is periodic with period $L$ or $L$-perodic, if we have $f(x +L) =f(x)$ for every real number $x$.
Example
$f(x) = \sin x$ and $g(x) = \cos x$ are $2 \pi $-periodic, and also $4 \pi $-periodic, $6 \pi$ -periodic , etc.
$f(x) = x$ is not periodic.
$f(x) = 1 $ is $L$-periodic for every $L$.
Remarks
If a function is $L$-periodic, then we have $f(x + kL) = f(x) $ for every integer $k$.
Definition: $\Z$-Periodic
If a function $f$ is $1$-periodic functions are sometimes also called $\Z$-periodic.
Example
$\forall n $, the functions
$$\cos (2 \pi n x), \sin (2 \pi n x), e^{2 \pi i n x}$$are all $\Z $-periodic.
Note
For simplicity, we shall only deal with functions which are $\Z $-periodic.