# Equivalence of two definitions of Fourier Series

I want to know why the following two definitions of Fourier series are equvalent:

1. $\displaystyle f(t)=\frac{a_0}{2}+\sum^{\infty}_{n=1}{(a_n\cos n\omega t+b_n\sin{n\omega t}})$

2. $\displaystyle f(t)=\frac{a_0}{2}+\sum^{\infty}_{n=1}{(a_n\cos nt+b_n\sin{n t}})$

Are these two definitions equivalent?

Thanks!

• In the first definition you can choose the period, so it is more flexible. – principal-ideal-domain Nov 24 '15 at 20:57

The second definition assumes a period of $2 \pi$ where as in the first definition $\omega= \frac{\pi}{L}$ where $2L$ is the period of the function, generally just stick to the first one as it's a more general definition and notice if you let $2L=2\pi$ you just get the second definition! I use $\displaystyle f(t)=\frac{a_0}{2}+\sum^{\infty}_{n=1}{(a_n\cos (\frac{n\pi}{L}t)+b_n\sin({\frac{n\pi}{L}t})})$ then there's no "ambiguity" among the variables.