(a) Suppose that $F(z), z=x+i y, x, y \in \mathbb{R}$, is analytic in the upper-half complex $z$-plane and $O(1 / z)$ as $z \rightarrow \infty, y \geqslant 0$. Show that the real and imaginary parts of $F(x)$, denoted by $U(x)$ and $V(x)$ respectively, satisfy the so-called Kramers-Kronig formulae:

$$U(x)=H V(x), \quad V(x)=-H U(x), \quad x \in \mathbb{R} .$$

Here, $H$ denotes the Hilbert transform, i.e.,

$$(H f)(x)=\frac{1}{\pi} \mathrm{PV} \int_{-\infty}^{\infty} \frac{f(\xi)}{\xi-x} d \xi$$

where $\mathrm{PV}$ denotes the principal value integral.

(b) Let the real function $u(x, y)$ satisfy the Laplace equation in the upper-half complex z-plane, i.e.,

$$\frac{\partial^{2} u(x, y)}{\partial x^{2}}+\frac{\partial^{2} u(x, y)}{\partial y^{2}}=0, \quad-\infty<x<\infty, \quad y>0$$

Assuming that $u(x, y)$ decays for large $|x|$ and for large $y$, show that $F=u_{z}$ is an analytic function for $\operatorname{Im} z>0, z=x+i y$. Then, find an expression for $u_{y}(x, 0)$ in terms of $u_{x}(x, 0)$.