Complex Fourier Series for f(t)=2sin(πt) with Periodicity

[twoscoops]

Homework Statement

Find the complex Fourier series of the periodic function

f(t)=2sin(πt) 0 < t < 1 and f(t+1) = f(t) for all t. (π is pi)

Homework Equations

http://upload.wikimedia.org/math/9/d/7/9d7f73fbcba87cbff485e66646aa541d.png
http://upload.wikimedia.org/math/5/2/8/52890b286b5e8481ee9d4d56f45081ac.png
http://upload.wikimedia.org/math/b/0/6/b06b197a31293ddd3a9b3812f419259d.png

The Attempt at a Solution

ive tried many times using the formula's and trying to rearrange to get exponentials but i always end up with something that looks really wrong, can someone point out any simplifications that can be made and do i need to integrate by parts (even though i can't see how that can be done). Any help will be greatly appriciated. Thanks

 

[Whitishcube]

I haven't worked with Fourier analysis or complex variables, but I would say to work with the first equation, the c_n. You can use integration by parts by treating i and n as constants and plugging in f(t) into the spot for f(x). That's where I would start, since after you find c_n the rest is cake.