Let the function $f(z)$ be analytic in the upper half-plane and such that $|f(z)| \rightarrow 0$ as $|z| \rightarrow \infty$. Show that

$$\mathcal{P} \int_{-\infty}^{\infty} \frac{f(x)}{x} d x=i \pi f(0),$$

where $\mathcal{P}$ denotes the Cauchy principal value.

Use the Cauchy integral theorem to show that

$$\mathcal{P} \int_{-\infty}^{\infty} \frac{u(x, 0)}{x-t} d x=-\pi v(t, 0), \quad \mathcal{P} \int_{-\infty}^{\infty} \frac{v(x, 0)}{x-t} d x=\pi u(t, 0),$$

where $u(x, y)$ and $v(x, y)$ are the real and imaginary parts of $f(z)$.