# The complex Fourier series

Recall the Fourier series expansion of a square wave, triangle wave, and sawtooth wave that we looked at before. That expansion described these periodic waveforms as sums of cosines, and showed the Fourier series coefficients*A*. We can equivalently describe them as sums of complex exponentials, where each cosine requires two complex exponentials (phasors rotating in opposite directions). Each complex exponential is weighted by a complex constant

_{k}*X*, representing both magnitude and phase. The following applet shows this expansion:

_{k}

Only the magnitude of each coefficient *X _{k} *is shown. The
key differences are that now there are frequency components shown at both
positive and negative frequencies. The negative frequencies correspond
to phasors rotating clockwise. The sum of a matching pair of
components will equal a cosine. Notice also that the amplitude of the components
is half that of the previous example, |

*X*| = |

_{k}*A*|/2. This is because there are now two components, one at negative frequencies and one at positive frequencies, that contribute.

_{k}Note that, as before, the right edge of each vertical red bar is aligned with the axis below, which is why the above plot looks like it is slightly off center.