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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.

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

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