Does a Fourier series always have period $2\pi$?

Question

If you do the Fourier series for $\sin{3x}$

The frequency is $\frac{\pi}{3}$ and the period is $2\pi$ ?

How does the Fourier series model $\sin{3x}$ ?

Answer 1

Fourier series have period $p=\frac{2\pi}{m}$ for $\sin(mx)$, so the period of $\sin(3x)$ is $\frac{2\pi}{3}$, ie, $p=\frac{2\pi}{3}$. The more general form of a Fourier series with period $p$ has the following representation: $$ S_R(x)= \frac{a_0}{2}+\sum^{R}_{\xi=1}a_{\xi}\cos\left(\frac{2\pi \xi x}{p}\right)+b_{\xi}\sin\left(\frac{2\pi \xi x}{p}\right) \\ =\sum_{|\xi|\leq R}c_{\xi}e^{\frac{2\pi i\xi\cdot x}{p}}. $$ Here, we have the integrals $$ a_0=\int^{x_0+p}_{x_0}f(x)dx,~ a_{\xi}=\frac{2}{p}\int^{x_0+p}_{x_0}f(x)\cos\left(\frac{2\pi\xi x}{p}\right)dx \\ b_{\xi}=\frac{2}{p}\int^{x_0+p}_{x_0}f(x)\sin\left(\frac{2\pi\xi x}{p}\right)dx,~ c_{\xi}=\frac{1}{p}\int^{x_0+p}_{x_0}f(x)e^{-\frac{2\pi i\xi x}{p}}dx \\ $$

So you will need to proceed as you would in the same way as you do for Fourier series with period $2\pi$ functions, but use this new adjusted formula. Can you do this yourself?