Fourier series - correspondence between complex and real coefficients.

[theneedtoknow]

Hello,

I know how to get the full Fourier series with complex coefficients and with real coefficients, and I know the relationship between An, Bn and Cn. However, I don't know why the relationship between them is what is it. Can someone either explain to me where the relationship comes from, or at least point out the steps I need to do in order to derive the relationship myself?

(This is not a homework problem, it's just that my book simply gives me the relationships between Cn, An and Bn, but they do not show how they derived them)

 

[mathman]

Usual complex Fourier series f(x)=ΣC_ne^inx, with n [-∞,∞]. To get series in terms of cos(nx) and sin(nx), first separate out the n=0 term (constant). Then combine the n term with the -n term, using the fact that cos(nx)={e^inx+e^-inx}/2 and sin(nx)={e^inx-e^-inx}/2i to combine C_n and C_-n To get A_n and B_n.

 

[theneedtoknow]

Thank you :)