# A claim regarding Fourier Series

## Summary:

Claim : "A periodic function $f(u)$ satisfying $\int_{0}^{1}f(u)du=0$ can generally expanded into a Fourier Series: $f(u)=\sum_{m=1}^{\infty} [a_m \sin{(2\pi mu)}+b_m\cos{(2\pi mu)}]$ "

## Main Question or Discussion Point

This is written on Greiner's Classical Mechanics when solving a Tautochrone problem.
Firstly,I don’t understand why we didn’t use the term $m=0$
and Sencondly, how the integrand helps us to fulfill the Dirichlet conditions. That means,how do we know that the period is 1?Thanks

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BvU
why we didn’t use the term $m=0$
$\int_{0}^{1}f(u)du=0$ makes $b_0=0$. The coefficient $a_0$ is arbitrary, so $m=0$ is better left out.
Periodicity is built in through $\sin\left(2\pi m (u+1) \right )$ $= \sin\left(2\pi m u \right )$ and idem $\cos$.