I'm trying to crack this problem but I can't seem to get it right.

It's the third exercise from from chapter 2 of "Complex Analysis" by Stein & Shakarchi.

I've drawn the sector (contour C) here (sorry for the crappy MS-paint picture):

As there are no poles within the sector I set $\displaystyle \oint_{C}f(z)dz = 0$, so that leaves

$\displaystyle 0 = \int_{0}^{A}e^{-Ax}dx + \int_{0}^{w}iAe^{-A^2e^{i\theta}}e^{i\theta}d\theta + \int_{0}^{A}e^{iw}e^{-Ate^{iw}}dt$

Now, I assume one of these three components goes to zero as A goes to infinity and indeed the second one (with $\displaystyle \theta$ as variable) does. Then I'm stuck because what I find $\displaystyle \int_{0}^{A = x = \infty}e^{-Ax}dx = 1/A$ doesn't lead me to anything remotely like the answer Wolfram's online calculator gave me if I use integration by parts

Can anyone help me with this problem, what am I doing wrong?