course | course_year | question_number | tags | title | year | |||
---|---|---|---|---|---|---|---|---|
Complex Analysis or Complex Methods |
IB |
10 |
|
Paper 1, Section II, A |
2018 |
(a) Let
$$\lim {R \rightarrow \infty} \oint{C} \frac{e^{i z^{2} / 4 \pi}}{e^{z / 2}-e^{-z / 2}} d z$$
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
for
[You may use Jordan's Lemma without proof.]