For a function $f \in L^2(\mathbb{T})$ (where $\mathbb{T}$ denotes the unit circle) I know that it can be expressed as $f(z) = \sum_{j = -\infty}^{\infty}f_j z^j$. The Fourier coefficients are given by $f_j = \langle f, z^j\rangle_{L^2(\mathbb{T})}$. My doubt is regarding finding the Fourier series expansion of a function $f$ which is defined as $$f(z)=\begin{cases}1\quad \text{if}\quad z= e^{i\theta},\: \theta \in [0,\pi]\\ 0\quad \text{if}\quad z= e^{i\theta},\: \theta \in (\pi,2\pi) \end{cases}$$
I have attempted to compute the Fourier coefficients of $f$ and found them as $ f_j = 0 \quad \text{if $j$ is even and} \quad \frac{1}{πij} \quad \text {if $j$ is even}$.
Thus, $f(z)= \sum_{j = -\infty}^{\infty}\frac{1}{πi(2j+1)}$. However the final answer doesn't seem right. But at the same time I don't know where am I doing a mistake.