Thanks a lot.
Another thing, what about the complex "version".
Is it correct what I did in the pic?
I'm not sure about the last part, can I say that f and g are actually the same?
And another thing (last one).
a_n and b_n also depends on x (cos(nx), sin(nx)). is changing only the outside cos(nx) and sin(nx) is enough? shouldn't I also place x+y inside of a_n and b_n instead of the x or maybe to recalculate them using the c_n?