$\displaystyle \|f\|^2 = \sum_{n=-\infty}^{\infty}|c_n|^2$ where $\displaystyle c_n$ are the fourier coefficients.

What does Parseval's identity look like if you use the coefficients from the real form of the fourier series, i.e. $\displaystyle a_n = c_n + c_{-n}$, $\displaystyle b_n = i(c_n - c_{-n})$ and $\displaystyle f \sim \frac{a_0}{2}+ \sum_{n=1}^{\infty}(a_n\cos(n\Omega t) + b_n\sin(n\Omega t) )$

I've seen it used like this, but I can't figure out how you get here: $\displaystyle \|f\|^2 = \left|\frac{a_0}{2}\right|^2 +\frac{1}{2}\sum_{n=1}^{\infty}(|a_n|^2+|b_n|^2)$