# Complex Fourier Series

1. Apr 30, 2012

### Tosh5457

Hi, I don't understand why does n goes from -∞ to +∞ in the complex Fourier series, but it goes from n=1 to n=+∞ in the real Fourier series?

2. Apr 30, 2012

### EWH

3. Apr 30, 2012

### HallsofIvy

Staff Emeritus
"Real Fourier Series" are in the form $\sum a_ncos(nx)+ b_nsin(nx)[tex] cosine is an even function and sine is an odd function so that if we did use negative values for n, it wouldn't give us anything new: [itex]a_{-n}cos(-nx)+ b_n sin(-nx)= a_{-n}cos(nx)- b_{-n} sin(nx)$ and would can then combine that with the corresponding "n" term: $( a_n+ a_{-n})cos(nx)+ (b_n- b_{-n})sin(nx)$

Another, but equivalent, way of looking at it is that $cos(nx)= (e^{inx}+ e^{-inx})/2$ and $sin(nx)= (e^{inx}+ e^{-inx})/2i$ so that sin(nx) and cosine(nx) with only positive n includes exponentials with both positive and negative n.

4. May 1, 2012

Thanks