(a) Let $f(z)$ be defined on the complex plane such that $z f(z) \rightarrow 0$ as $|z| \rightarrow \infty$ and $f(z)$ is analytic on an open set containing $\operatorname{Im}(z) \geqslant-c$, where $c$ is a positive real constant.

Let $C_{1}$ be the horizontal contour running from $-\infty-i c$ to $+\infty-i c$ and let

$$F(\lambda)=\frac{1}{2 \pi i} \int_{C_{1}} \frac{f(z)}{z-\lambda} d z$$

By evaluating the integral, show that $F(\lambda)$ is analytic for $\operatorname{Im}(\lambda)>-c$.

(b) Let $g(z)$ be defined on the complex plane such that $z g(z) \rightarrow 0$ as $|z| \rightarrow \infty$ with $\operatorname{Im}(z) \geqslant-c$. Suppose $g(z)$ is analytic at all points except $z=\alpha_{+}$and $z=\alpha_{-}$which are simple poles with $\operatorname{Im}\left(\alpha_{+}\right)>c$ and $\operatorname{Im}\left(\alpha_{-}\right)<-c$.

Let $C_{2}$ be the horizontal contour running from $-\infty+i c$ to $+\infty+i c$, and let

$$\begin{gathered} H(\lambda)=\frac{1}{2 \pi i} \int_{C_{1}} \frac{g(z)}{z-\lambda} d z \\ J(\lambda)=-\frac{1}{2 \pi i} \int_{C_{2}} \frac{g(z)}{z-\lambda} d z . \end{gathered}$$

(i) Show that $H(\lambda)$ is analytic for $\operatorname{Im}(\lambda)>-c$.

(ii) Show that $J(\lambda)$ is analytic for $\operatorname{Im}(\lambda)<c$.

(iii) Show that if $-c<\operatorname{Im}(\lambda)<c$ then $H(\lambda)+J(\lambda)=g(\lambda)$.

[You should be careful to make sure you consider all points in the required regions.]