The arc is the following integral (this arc is part of a semicircle , where the bottom part is over the real axis, and it traverses in a CCW orientation)
$$ \int_0^\pi {re^{ia}ire^{ia}\over e^{r\cos(a)}e^{ir\sin(a)}}\,da $$
From most examples I have seen the integral vanishes as $ r\rightarrow \infty,$ but how can I prove this? I have looked up Jordan's Lemma and also have seen people use the triangle inequality.
Thank you very much for your time and help.