# 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?

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You seem to already have given two great reasons: it's often cleaner and more efficient to work with exponentials. What else do you want to hear? –  Qiaochu Yuan Sep 21 '12 at 2:45
I wanted to know if there was a mathematical reason to use one form over the other. Something beyond personal preference. –  Michael Sep 21 '12 at 5:05
Cleanliness and efficiency aren't mathematical reasons? –  Qiaochu Yuan Sep 21 '12 at 5:21
No, cleanliness and efficiency (in this context with no measurable baseline) are subjective in my opinion. One may find something efficient, while another may find it cumbersome. –  Michael Sep 24 '12 at 23:07

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?

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