$$I=\oint_{C} \frac{e^{i z^{2} / \pi}}{1+e^{-2 z}} d z$$

where $C$ is the rectangle with vertices at $\pm R$ and $\pm R+i \pi$, traversed anti-clockwise.

(i) Show that $I=\frac{\pi(1+i)}{\sqrt{2}}$.

(ii) Assuming that the contribution to $I$ from the vertical sides of the rectangle is negligible in the limit $R \rightarrow \infty$, show that

$$\int_{-\infty}^{\infty} e^{i x^{2} / \pi} d x=\frac{\pi(1+i)}{\sqrt{2}}$$

(iii) Justify briefly the assumption that the contribution to $I$ from the vertical sides of the rectangle is negligible in the limit $R \rightarrow \infty$.