Complex Fourier Series?

ols

I need to find the coefficients in the complex fourier series for the fucntion f(t) = cos(1/2pi*t) for |t|< 1 and f(t) = 0 for 1<|t|< T/2, where T= period.

Also, I need to find c_n as a function of w_n=(2pi)/T and T, by first writing cos(1/2pi*t) as complex exponentials and performing integration.

Completely Stuck

Any help would be greately appreciated

kb

c_n = ∫(t = -1 to 1) e^(-iπnt/T) cos(πt/2) dt

......= ∫(t = -1 to 1) e^(-iπnt/T) * (1/2) [e^(iπt/2) + e^(-iπt/2)] dt, by complex def. of cosine

......= (1/2) ∫(t = -1 to 1) [e^(iπt (1/2 - n/T)) + e^(-iπt (1/2 + n/T))] dt

......= (1/2) [e^(iπt (1/2 - n/T))/(iπ (1/2 - n/T)) + e^(-iπt (1/2 + n/T))/(-iπ (1/2 + n/T))] {for t = -1 to 1}

......= (1/2) [e^(iπ (1/2 - n/T))/(iπ (1/2 - n/T)) + e^(-iπ (1/2 + n/T))/(-iπ (1/2 + n/T))]

- (1/2) [e^(-iπ (1/2 - n/T))/(iπ (1/2 - n/T)) + e^(iπ (1/2 + n/T))/(-iπ (1/2 + n/T))].

Note: If 1/2 - n/T or 1/2 + n/T equal 0, then there may be an exceptional case or 2; in such a case, compute that particular coefficient separately.

I hope this helps!