Perceval Theorem in even and odd function

Question

I would like to understand this example: the Parseval's Theorem is given by:

\[\frac{1}{L} \int_{-L}^{L} (f(x))^2 dx = \frac{a_0^2}{2} + \sum_{n=1}^{\infty} a_n^2 + b_n^2\] or \[\frac{1}{L} \int_{-L}^{L} (f(x))^2 dx = 2{a_0^2} + \sum_{n=1}^{\infty} a_n^2 + b_n^2\] What is the Parseval's Theorem if the function is even or odd?

Answer 1

It helps a lot to write down the expressions for the Fourier coefficients, \begin{align*}& {a_0} = \frac{1}{{2L}}\int_{{\, - L}}^{L}{{f\left( x \right)\,\mathrm{d}x}}\\ & {a_n} = \frac{1}{L}\int_{{\, - L}}^{L}{{f\left( x \right)\cos \left( {\frac{{n\,\pi x}}{L}} \right)\,\mathrm{d}x}}\hspace{0.25in}\hspace{0.25in}n = 1,2,3, \ldots \\ & {b_n} = \frac{1}{L}\int_{{\, - L}}^{L}{{f\left( x \right)\sin \left( {\frac{{n\,\pi x}}{L}} \right)\,\mathrm{d}x}}\hspace{0.25in}\hspace{0.25in}n = 1,2,3, \ldots.\end{align*}

Considering the $a_0$ integral, for instance, we can see that if $f(x)$ were odd, $a_0$ would be zero (the negative area cancels the positive area). You can make similar observations for the other two cases to see when $a_n$ and $b_n$ vanish.