The Hilbert transform $\hat{f}$ of a function $f$ is defined by

$$\hat{f}(x)=\frac{1}{\pi} P \int_{-\infty}^{\infty} \frac{f(y)}{y-x} d y$$

where $P$ denotes the Cauchy principal value.

(i) Compute the Hilbert transform of $(1-\cos t) / t$.

(ii) Solve the following Riemann-Hilbert problem: Find $f^{+}(z)$ and $f^{-}(z)$, which are analytic functions in the upper and lower half $z$-planes respectively, such that

$$\begin{aligned} &f^{+}(x)-f^{-}(x)=\frac{1-\cos x}{x}, \quad x \in \mathbb{R} \\ &f^{\pm}(z)=O\left(\frac{1}{z}\right), \quad z \rightarrow \infty, \quad \operatorname{Im} z \neq 0 \end{aligned}$$