Fourier Series/Pi Squared minus x Squared over Minus Pi to Pi

Theorem
Let $f \left({x}\right)$ be the real function defined on $\left({-\pi \,.\,.\, \pi}\right)$ as:


 * $f \left({x}\right) = \pi^2 - x^2$

Then its Fourier series can be expressed as:

Proof
We have that:
 * $\pi^2 - \left({-x}\right)^2 = \pi^2 - x^2$

and so $f \left({x}\right)$ is even on $\left({-\pi \,.\,.\, \pi}\right)$.

It follows from Fourier Series for Even Function over Symmetric Range:


 * $\displaystyle f \left({x}\right) \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty a_n \cos n x$

where for all $n \in \Z_{> 0}$:


 * $a_n = \displaystyle \frac 2 \pi \int_0^\pi f \left({x}\right) \cos n x \rd x$

Thus by definition of $f$:

Then for $n > 0$:

Splitting this up into two:

Finally: