Suppose that the real function $u(x, y)$ satisfies Laplace's equation in the upper half complex $z$-plane, $z=x+i y, x \in \mathbb{R}, y>0$, where

$$u(x, y) \rightarrow 0 \quad \text { as } \quad \sqrt{x^{2}+y^{2}} \rightarrow \infty, \quad u(x, 0)=g(x), \quad x \in \mathbb{R} .$$

The function $u(x, y)$ can then be expressed in terms of the Poisson integral

$$u(x, y)=\frac{1}{\pi} \int_{-\infty}^{\infty} \frac{y g(\xi)}{(x-\xi)^{2}+y^{2}} d \xi, \quad x \in \mathbb{R}, y>0$$

By employing the formula

$$f(z)=2 u\left(\frac{z+\bar{a}}{2}, \frac{z-\bar{a}}{2 i}\right)-\overline{f(a)}$$

where $a$ is a complex constant with $\operatorname{Im} a>0$, show that the analytic function whose real part is $u(x, y)$ is given by

$$f(z)=\frac{1}{i \pi} \int_{-\infty}^{\infty} \frac{g(\xi)}{\xi-z} d \xi+i c, \quad \operatorname{Im} z>0$$

where $c$ is a real constant.