Three Fourier series: sines, cosines, and exponentials \( e^{ikx} \)
\( \because S(x) = \sum_{n=1}^{\infty} b_n \sin(n x), \quad F(x) = \sum_{n=-\infty}^{\infty} c_n e^{i n x} \)
\( \because \int_0^\pi \sin(n x) \sin(m x) dx = 0, \quad \text{for } n \neq m \)
\( \therefore \text{Sinusoids and exponentials form an orthogonal basis for representing wave-like solutions} \)
\( \because F(k) = \int_{-\infty}^{\infty} f(x) e^{-i k x} dx \)
\( \therefore f(x) = \int_{-\infty}^{\infty} F(k) e^{i k x} dk \)
\( \because u(x, t) = A(x) B(t) \quad \text{(separation of variables)} \)
\( \because A(x) \text{ and } B(t) \text{ satisfy wave equations} \)
\( \therefore A(x) = e^{i k x}, \quad B(t) = e^{-i \omega t} \)
\( \because \text{Fourier transform solutions are plane waves in space and time} \)
\( \therefore u(x, t) = \hat{u}(y) e^{i (k x - \omega t)} \)