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So, if we represent a function of time using Fourier series: $$f(t)=\frac{a_0}{2}+\sum_{n=1}^{\infty}\bigg[{a_n\cos{(n\omega_0t)}+b_n\sin{(n\omega_0t)}}\bigg]$$ what is the point of that halved first coefficient? Couldn't it simply be written as: $$f(t)=a_0+\sum_{n=1}^{\infty}\bigg[{a_n\cos{(n\omega_0 t)}+b_n\sin{(n\omega_0 t)}}\bigg]$$

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It is done so that the same formula can be used for all values of $n$: $$ a_n=\frac{1}{\pi}\int_0^{2\pi}f(t)\cos(n\,t)\,dt,\quad n\ge0. $$ Otherwise, the definition of $a_0$ should be $$ a_0=\frac{1}{2\,\pi}\int_0^{2\pi}f(t)\,dt. $$

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