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 $|z| < 1$, $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Let $\varphi \in \mathbb{R}$. Show that, for any complex number $z$ such that $|z| < 1$, $g(z) = \frac{1}{2\pi} \int_{\varphi}^{\varphi + 2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t$.
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 $|z| < 1$,
$$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$
Let $\varphi \in \mathbb{R}$. Show that, for any complex number $z$ such that $|z| < 1$, $g(z) = \frac{1}{2\pi} \int_{\varphi}^{\varphi + 2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t$.