I am really unsure as to how to tackle this contour integral question, Can I get a comprehensive guide to tackling this question?
$$H(\lambda)= \oint_C \frac{e^{i\lambda z^2}}{z-2-i} \space dz$$ where the contour C comprises the straight line segment $L_1$ given by $y = 0$ and $1 ≤ x ≤ R$, the straight line segment $L_2$ given by $x = R$ and $0 ≤ y ≤$ √ $(R2 − 1)$, and the steepest descent curve$ L_3$ for the function $p(z) = iz^2$ emanating from $x = 1$ and $0 ≤ y ≤ $√ $(R2 − 1)$
Find the value of $H(\lambda)$ (You may assume that $R>2$).
Within the region enclosed by the contour $C$, the integrand is analytic except at the simple pole $z=2+i$. Thus, the integral may be easily evaluated by the residue theorem as
$$H(\lambda) = i 2 \pi e^{i \lambda (2+i)^2} = 2 \pi \, e^{-4 \lambda} \left ( -\sin{3 \lambda} + i \cos{3 \lambda}\right ) $$