Define

$$F^{\pm}(x)=\lim _{\epsilon \rightarrow 0} \frac{1}{2 \pi i} \int_{-\infty}^{\infty} \frac{f(t)}{t-(x \pm i \epsilon)} d t, \quad x \in \mathbb{R}$$

Using the fact that

$$F^{\pm}(x)=\pm \frac{f(x)}{2}+\frac{1}{2 \pi i} P \int_{-\infty}^{\infty} \frac{f(t)}{t-x} d t, \quad x \in \mathbb{R}$$

where $P$ denotes the Cauchy principal value, find two complex-valued functions $F^{+}(z)$ and $F^{-}(z)$ which satisfy the following conditions

1. $F^{+}(z)$ and $F^{-}(z)$ are analytic for $\operatorname{Im} z>0$ and $\operatorname{Im} z<0$ respectively, $z=x+i y$;

2. $F^{+}(x)-F^{-}(x)=\frac{\sin x}{x}, \quad x \in \mathbb{R}$;

3. $F^{\pm}(z)=\mathrm{O}\left(\frac{1}{z}\right), \quad z \rightarrow \infty, \quad \operatorname{Im} z \neq 0$.