A claim regarding Fourier Series

  • #1

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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Answers and Replies

  • #2
BvU
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Hi,
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##.
 

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