An orthonormal and complete basis for functions \(g(\theta)\) that are periodic in \(\theta \in[0,2 \pi)\), where \(g(\theta)=g(\theta+2 \pi)\), are the imaginary exponentials

\[\begin{equation*}f_{n}(\theta)=\frac{e^{i n \theta}}{\sqrt{2 \pi}} \tag{2.103}\end{equation*}\]where \(n \in \mathbb{Z}\), an integer. In this problem, we will study this space.

(a) First show that these basis functions are orthonormal on \(\theta \in[0,2 \pi)\).

(b) Consider the derivative operator on this space:

\[\begin{equation*}\hat{D}=-i \frac{d}{d \theta} \tag{2.104}\end{equation*}\]

What are its eigenvalues and eigenvectors?

(c) An equivalent way to express this space is to use a different basis in which sines and cosines are the basis elements:

\[\begin{equation*}c_{n}(\theta)=\frac{1}{\sqrt{\pi}} \cos (n \theta), \quad s_{n(\theta)=\frac{1}{\sqrt{\pi}} \sin (n \theta), \tag{2.105}\end{equation*}\]
for \(n eq 0\). Show that these basis functions are also orthonormal on \(\theta \in\) \([0,2 \pi)\). What is the normalized basis element for \(n=0\) ?

(d) In this sine/cosine basis, determine the matrix elements of the derivative operator defined in part (b). Can you explicitly write down the form of the derivative operator as a matrix in this basis?