Solving Integral(s*exp(isr)/(s^2-k^2)) ds

Greetings,

I'm trying to teach myself contour integration/complex analysis and finding it a little more challenging than I expected. To make a long story short, it's not obvious to me how to make the semi-circle go to 0 in the following: Integral(s*exp(isr)/(s^2-k^2)) ds from -inf to inf.

The contribution from the semi-circle is given by: integral(i*R^2*exp

(2*i*theta)*exp(i*r*R*exp(i*theta))/(R^2*exp(2*i*theta) - k^2)) from 0 to pi. As R -> infinity, this is supposed to go to 0. I don't see this.

To arrive at the equation I mentioned above, I substitute R*exp(i*theta) in for s above:

integral(R*exp(i*theta)*exp(i*r*R*exp(i*theta))*i* R*exp(i*theta)/R^2*exp(2*i*theta) - k^2)) dtheta from 0 to pi

= integral(i*R^2*exp(2*i*theta)*exp(i*r*R*exp(i*thet a))/(R^2*exp(2*i*theta) - k^2)) dtheta from 0 to pi

At this point, I want to try to get it to go to 0:

Dividing top and bottom by R^2*exp(2*i*theta):

integral(i*exp(i*r*R*exp(i*theta))/(1 - (k^2 / R^2*exp(2*i*theta))))from 0 to pi

Taking R to infinity:

integral(i*exp(i*r*R*exp(i*theta))/(1 - 0)) from 0 to pi

which finally gives:

integral(i*exp(i*r*R*exp(i*theta))) from 0 to pi, R -> inf

...which doesn't obviously go to 0 as far as I can tell.

Any help appreciated!

mapsread

P.S. I can follow many arguments for seeing the integral go to 0, but the extra R in the numerator is what's giving me fits. Thanks!