complex exponential Fourier series
/ FOOR-ee-ay /
Sines and cosines are really two faces of a single, simpler object — the complex exponential e^{i theta}, which by Euler's formula spins as cos theta + i sin theta. The complex exponential Fourier series exploits that to write the whole expansion in one tidy line instead of two parallel sums of cosines and sines. Think of it as repackaging the harmonic content of a signal as a single list of complex numbers, one per frequency, where the size of each number is the strength of that harmonic and its angle is the phase.
For period 2L the series reads f(x) = sum over n from -infinity to infinity of c_n e^{i n pi x / L}, where the complex coefficients are c_n = (1/(2L)) integral over one period of f(x) e^{-i n pi x / L} dx. Now n runs over all integers, positive and negative; the negative frequencies are not mystical — they are exactly what is needed so that a real signal's terms pair up (c_{-n} is the complex conjugate of c_n) and the imaginary parts cancel, leaving a real sum. The link back to the real form is direct: c_n = (a_n - i b_n)/2 for n > 0, c_0 = a_0/2. One coefficient formula now does the job of two, because e^{-i n pi x / L} carries the cosine and the sine together.
This is the form physicists and engineers actually compute with. It makes the algebra of shifts, derivatives, and convolutions almost trivial (differentiating just multiplies c_n by i n pi / L), it generalizes seamlessly to the Fourier transform when the period grows without bound, and it is the natural language for the discrete and fast Fourier transform that runs inside every digital audio and image pipeline. The only conceptual toll is comfort with complex numbers and the slightly odd-looking negative-frequency terms.
For f(x) = e^{x} on (-pi, pi), period 2pi: c_n = (1/(2pi)) integral from -pi to pi of e^{x} e^{-i n x} dx = sinh(pi) / pi * (-1)^n / (1 - i n) — a single complex formula covering every harmonic, positive and negative n alike.
One exponential integral, trivially done, replaces the two real integrals for a_n and b_n.
The minus sign in e^{-i n pi x / L} inside the coefficient integral is not optional: it is the conjugate of the basis function, and flipping it (or dropping the 1/(2L)) silently corrupts every coefficient's phase.