When to use Exponential Fourier series versus Trigonometric Fourier series? I am usually more comfortable working with sines and cosines. However, its often cleaner and more efficient to work with exponentials.
Is there a reason other then preference, to use one over the other?
EDIT:
Thinking more on this, Exp Fourier series allows the spectra to be expanded into negative frequency. But in the end, isnt this the same information?
 A: Speaking as an EE, imaginary exponentials are generally easier to manipulate then their trigonometric counterparts since they essentially let you get a polynomial out of your sin/cos function.
For example if I gave you
$$
\cos \Bigg(n_1\omega t+\theta_1\Bigg)\cos\Bigg(n_2\omega t+\theta_2\Bigg)
$$
You can imagine trying to do any kind of algebra with that. Using exponentials we can write it in a simpler form.
$$
\Bigg[\mathrm e^{\theta_1}\mathrm e^{jn_1\omega t}+\mathrm e^{\theta_1}\mathrm e^{-jn_1\omega t} \Bigg] \Bigg[\mathrm e^{\theta_2}\mathrm e^{jn_2\omega t}+\mathrm e^{\theta_2}\mathrm e^{-jn_2\omega t} \Bigg]
$$
Do some math and you get him down to
$$
\Bigg[\mathrm e^{\theta_1\theta_2}\mathrm e^{j\omega t(n_1+n_2)}+\mathrm e^{\theta_1\theta_2}\mathrm e^{-j\omega t(n_1+n_2)} \Bigg]+\Bigg[\mathrm e^{\theta_1\theta_2}\mathrm e^{j\omega t(n_2-n_1)}+\mathrm e^{\theta_1\theta_2}\mathrm e^{-j\omega t(n_2-n_1)} \Bigg]
$$
And simplify to
$$
\cos \Bigg[(n_1+n_2)\omega t + \theta_1\theta_2) \Bigg]+\cos \Bigg[(n_2-n_1)\omega t + \theta_1\theta_2) \Bigg]
$$
That looks much nicer doesn't it?
