The Complex Fourier Series

The sine-and-cosine Fourier series carries two families of coefficients and a special constant term — a little clumsy to push through calculations. Euler's formula e^{i\theta} = \cos\theta + i\sin\theta folds both families into one tidy package of complex exponentials.

Writing \cos\theta = \tfrac12(e^{i\theta} + e^{-i\theta}) and \sin\theta = \tfrac1{2i}(e^{i\theta} - e^{-i\theta}) and collecting terms, the whole series becomes a single sum over all integers — positive and negative:

f(x) = \sum_{n=-\infty}^{\infty} c_n\, e^{i n\pi x / L}, \qquad c_n = \frac{1}{2L}\int_{-L}^{L} f(x)\, e^{-i n\pi x / L}\,dx.

One coefficient, one formula

This is the same idea as before — projection onto orthogonal building blocks — but now the blocks are the exponentials e^{i n\pi x/L}, which are orthogonal under the complex inner product \int_{-L}^{L} f\,\overline{g}\,dx. There is just one formula for one family of coefficients c_n, and the negative-n terms carry the rest of the information.

The dictionary back to the real coefficients is short:

c_0 = \frac{a_0}{2}, \qquad c_n = \frac{a_n - i b_n}{2}, \qquad c_{-n} = \frac{a_n + i b_n}{2} = \overline{c_n}.

For a real function the coefficients come in conjugate pairs, c_{-n} = \overline{c_n}, and |c_n| is the strength of frequency n — the function's spectrum.

Complex exponentials are eigenfunctions of differentiation: \frac{d}{dx}e^{ikx} = ik\,e^{ikx}. Differentiating a complex Fourier series just multiplies each c_n by i n\pi/L — no product rule, no sign-juggling between sines and cosines. That single fact makes the exponential form the working language of signal processing and quantum mechanics, and it sets up the Fourier transform cleanly.