Trignometric Polynomial complex form

[phillyj]

Hi,
I'm trying to learn Fourier transforms by myself. I'm a bit confused about how the trignometric polynomial complex form was derived. I'm referring to this:

http://en.wikipedia.org/wiki/Trigonometric_polynomial

Now, I haven't taken complex analysis so I only know the basics. I used Euler's formula and got that far but I'm not sure what to do next.

Thanks

 

[micromass]

Substitute

$$\cos(nx)=\frac{e^{inx}+e^{-inx}}{2}, ~\sin(nx)=\frac{e^{inx}-e^{-inx}}{2i}$$

in the series and rearrange everything. This should give you the required form.