I am looking into Fourier series expansion of function that is non periodic on 0< x< p. This is under the tittle of "Half range expansion". All the books only talked about even or odd extension of f(x) to -p <x< p using either Fouries cosine series expansion or sine series expansion. But never using both for extension. If f(x) has both sine and cosine series expansion, it is possible to expand with both sine and cosine. the difference is with both expansion, f(x) = 0 for -p<x<0, but still the expansion is valid for 0<x<p. To me this is more closer to the original f(x)........ Instead of odd of even extension where the part in -p<x<0 is totally different shape from the f(x). To me I rather just null out the part -p<x<0. Can anyone tell me a better reason?