In Fourier Series when is it acceptable to just integrate half of period and double the result later to find coefficient?

Question

in finding the coefficient of Fourier Series, $a_0, a_n, b_n$. We integrate the periodic function $f(t)$ over the period $T$. That is

$$\frac1T\int^T_0 f(t)\ dt$$ $$\frac2T\int^T_0 f(t)\cos(n\omega t)\ dt$$ $$\frac2T\int^T_0 f(t)\sin(n\omega t)\ dt$$

However, for some function, $a_n$ and $b_n$ can be calculate by just integrate over half of period and multiply the result by 2 that means

$$\frac4T\int^{\frac T2}_0 f(t)\cos(n\omega t)\ dt$$ $$\frac4T\int^{\frac T2}_0 f(t)\sin(n\omega t)\ dt$$

However, I don't know if this approach can be used in all situation or just some special cases. If it can be used only in some special cases, then what is the requirement to use it.

Moreover, for some periodic function having the same positive and negative part for example the sinusoid. Does $a_0$ of this function always return the result of zero?

Answer 1

This can be done when $f$ has some symmetry around $T/2$, the midpoint of the interval $[0,T]$.

If $f(t)=f(T-t)$, then $$\begin{align} \int^T_0 f(t)\cos(n\,\omega\,t)\,dt&=\int^{T/2}_0 f(t)\cos(n\,\omega\,t)\,dt+\int^T_{T/2} f(t)\cos(n\,\omega\,t)\,dt\\ &=\int^{T/2}_0 f(t)\cos(n\,\omega\,t)\,dt+\int^{T/2}_{0} f(T-t)\cos(n\,\omega\,(T-t))\,dt\\ &=2\int^{T/2}_0 f(t)\cos(n\,\omega\,t)\,dt. \end{align}$$ A similar argument shows that $b_n=0$.

I leave to you the case when $f(T-t)=-f(t)$.