(a) Let $C$ be a rectangular contour with vertices at $\pm R+\pi i$ and $\pm R-\pi i$ for some $R>0$ taken in the anticlockwise direction. By considering

$$\lim _{R \rightarrow \infty} \oint_{C} \frac{e^{i z^{2} / 4 \pi}}{e^{z / 2}-e^{-z / 2}} d z$$

show that

$$\lim _{R \rightarrow \infty} \int_{-R}^{R} e^{i x^{2} / 4 \pi} d x=2 \pi e^{\pi i / 4}$$

(b) By using a semi-circular contour in the upper half plane, calculate

$$\int_{0}^{\infty} \frac{x \sin (\pi x)}{x^{2}+a^{2}} d x$$

for $a>0$.

[You may use Jordan's Lemma without proof.]