# How can i determine if the arc of the following contour integral vanishes as R approaches infinity?

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.

• Try showing that this is $$\oint_{|z|=r}\frac{z\,\mathrm{d}z}{e^z}$$ – robjohn Oct 2 '18 at 20:07
• How can an arc be an integral? – zhw. Oct 2 '18 at 22:30
• contour integral , over the complex plane integrals are paths taking from point a to b. – Victor Orta Oct 2 '18 at 23:47