Quote by matematikuvol
Thanks mate. Sorry again. Is there some easy way to see that that integral will go to zero? When you see
[tex]\oint\frac{e^{ikz}}{z}[/tex]?

Ok, that one "looks" like you're just going around the origin but really you mean the halfdisc contour in either halfplane. Around the origin, the integral is 2pi i. Otherwise if it's around the discs. You could be more specific like:
[tex]\mathop\oint\limits_{z=1} \frac{e^{ikz}}{z}dz[/tex]
that's really clear or in the other case:
[tex]\mathop\oint\limits_{D} \frac{e^{ikz}}{z}dz[/tex]
then clearly specify in the text what D is. In regards to you question about the integral over the upper halfdisc around the semicircle, well, you just need to plug it all in and analyze it to see what happens as R goes to infinity.