Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Show that, for all $\delta \in ]0, \pi[$ and all real $\varphi$, $$\int_{\varphi+\delta}^{\varphi+2\pi-\delta} \mathcal{P}(t,z) \, \mathrm{d}t \xrightarrow[z \rightarrow \mathrm{e}^{\mathrm{i}\varphi}]{} 0$$
Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$,
$$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$
Show that, for all $\delta \in ]0, \pi[$ and all real $\varphi$,
$$\int_{\varphi+\delta}^{\varphi+2\pi-\delta} \mathcal{P}(t,z) \, \mathrm{d}t \xrightarrow[z \rightarrow \mathrm{e}^{\mathrm{i}\varphi}]{} 0$$