I've seen lots of examples of periodic functions, but they all have one thing in common: They all involve at least one trigonometric term (e.g. $\sin\theta$, $\cos\theta$, etc.). My question is simple: are there any continuous, differentiable periodic functions that do not involve trigonometric terms? If not, why?
Trigonometry – Non-Trigonometric Continuous Periodic Functions
periodic functionstrigonometry
Best Answer
The simplest infinitely differentiable non-trigonometric* function I can think of is $$f(x)=\sum_{n\in\mathbb Z} e^{-(x-n)^2}\tag{1}$$ Periodicity is clear; differentiability follows from the fact that every derivative of $e^{-x^2}$ is of the form $p(x)e^{-x^2}$ for some polynomial $p$, and the series $$\sum_{n\in\mathbb Z} |p(x-n)| e^{-(x-n)^2}$$ converges uniformly on every bounded interval.
The function (1) is sometimes called the periodized Gaussian, although it seems that the same term is used for the nondifferentiable functions obtained by taking a central piece of Gaussian curve and repeating it.
(*) Not-explicitly-trigonometric. As others said, there is always a trigonometric series lurking in background.