(a) Functions $g_{1}(z)$ and $g_{2}(z)$ are analytic in a connected open set $\mathcal{D} \subseteq \mathbb{C}$ with $g_{1}=g_{2}$ in a non-empty open subset $\tilde{\mathcal{D}} \subset \mathcal{D}$. State the identity theorem.

(b) Let $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ be connected open sets with $\mathcal{D}_{1} \cap \mathcal{D}_{2} \neq \emptyset$. Functions $f_{1}(z)$ and $f_{2}(z)$ are analytic on $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ respectively with $f_{1}=f_{2}$ on $\mathcal{D}_{1} \cap \mathcal{D}_{2}$. Explain briefly what is meant by analytic continuation of $f_{1}$ and use part (a) to prove that analytic continuation to $\mathcal{D}_{2}$ is unique.

(c) The function $F(z)$ is defined by

$$F(z)=\int_{-\infty}^{\infty} \frac{e^{i t}}{(t-z)^{n}} d t$$

where $\operatorname{Im} z>0$ and $n$ is a positive integer. Use the method of contour deformation to construct the analytic continuation of $F(z)$ into $\operatorname{Im} z \leqslant 0$.

(d) The function $G(z)$ is defined by

$$G(z)=\int_{-\infty}^{\infty} \frac{e^{i t}}{(t-z)^{n}} d t$$

where $\operatorname{Im} z \neq 0$ and $n$ is a positive integer. Prove that $G(z)$ experiences a discontinuity when $z$ crosses the real axis. Determine the value of this discontinuity. Hence, explain why $G(z)$ cannot be used as an analytic continuation of $F(z)$.