Orthogonality of complex exponentials

We now show that the above defined periodic complex exponential constitute an orthonormal set of functions.

For this purpose we recall the definition of the complex inner product for two functions on an arbitrary interval [\(a,b\)], which is defined as:

$$ \langle f|g\rangle =\int_{a}^b \mathrm{d}x \ f^{ * } (x)g(x) $$

For two complex exponentials with indices \(m\) and \(n\) we have:

$$ \langle\varphi_{m}|\varphi_{n}\rangle=\int_{0}^{L} \mathrm{d}x \exp \left( \mathrm{i} \frac{2\pi}{L}(n-m)x \right) = \begin{cases} L & m=n \\ 0 & m \ne n \end{cases} $$

This can be compactly represented by introducing the Kronecker delta symbol:

$$\langle\varphi_{m}|\varphi_{n}\rangle=L\delta_{mn}$$

Thus, we now have a discrete set of orthonormal functions that can serve as a basis for representing arbitrary periodic function! The functions \(\varphi_{n}(x)\) are also called trigonometric polynomials since they can be expressed as:

$$\varphi_{n}(z)=z^{n}$$

Here \(z\) is a complex variable defined as:

$$z=\exp\left(\mathrm{i}\frac{2\pi x}{L}\right)$$