The complex Fourier series is defined over the interval from -∞ to +∞, while the regular Fourier series is defined from 0 to +∞. This distinction arises because the complex Fourier series incorporates both even and odd functions, specifically through the use of the complex exponential function, ei(nx) = cos(nx) + i sin(nx). The relationship between the coefficients of the complex and regular Fourier series reveals a factor of two difference, attributed to the inclusion of negative values of n in the complex series, which are already accounted for in the coefficients.
PREREQUISITESMathematicians, engineers, and students studying signal processing or harmonic analysis will benefit from this discussion, particularly those interested in the differences between complex and regular Fourier series.