This Topic investigates in detail the relationships between the trigonometric Fourier coefficients, and the complex Fourier coefficients. The relationships are demonstrated through the example of comparing the Fourier coefficients for the square wave periodic function. The Topic then moves onto the more general discussion of the properties of the complex Fourier coefficients, primarily conjugate symmetry resulting in symmetric amplitude and asymmetric phase for real valued signals. The Topic then moves onto the idea of plotting the complex Fourier coefficients as a function of the coefficient index, which is the basis for plotting the spectrum of the signal, as shown in future Topics. Since the Fourier coefficients are, in general, complex, then it becomes necessary to plot magnitude and phase on separate diagrams. Putting together this theory, the Topic then poses an example to consider, followed by a physical interpretation of the complex Fourier series.