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

Theorem
Let $\map f x$ be the real function defined on $\openint {-\pi} \pi$ as:


 * Sneddon-1-Exercise-6.png


 * $\map f x = \pi^2 - x^2$

$f$ can be expressed as a half-range Fourier cosine series thus:

Proof
We have that:
 * $\pi^2 - \paren {-x}^2 = \pi^2 - x^2$

and so $\map f x$ is even on $\openint {-\pi} \pi$.

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


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

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


 * $a_n = \displaystyle \frac 2 \pi \int_0^\pi \map f x \cos n x \rd x$

Thus by definition of $f$:

Then for $n > 0$:

Splitting this up into two:

Finally: