1. ## Maximum Likelihood estimate

I've been having problem with toy contour integration because most of the time I wasn't able to prove the integral around the arc length, sometimes the integral along the arc length of a semicircle is equal to 0.
I'm integrating $\int_{-\infty}^{\infty}\frac{cosx}{x^2+a^2}dx$. I follow what I've learned, so I integrate this over semicircle C with radius R. I found the poles, and use the residue theorem to obtain the integral. But I don't know how to show that the integral along this semicircle is 0 as R approaches infinity. There are two ways that I've seen my professor to do this is to use the inequality $\mid \int_{\gamma}f(z)dz \mid \leq sup \mid f(z) \mid length(\gamma)$. Sometimes, he uses the maximum likelihood estimate which I really have no idea about. Can someone help me on this? Are these two techniques always applicable and interchangable in this case and in some other similar integrals over some arc length. I really appreciate any help.

2. Originally Posted by namelessguy
I've been having problem with toy contour integration because most of the time I wasn't able to prove the integral around the arc length, sometimes the integral along the arc length of a semicircle is equal to 0.
I'm integrating $\int_{-\infty}^{\infty}\frac{cosx}{x^2+a^2}dx$. I follow what I've learned, so I integrate this over semicircle C with radius R. I found the poles, and use the residue theorem to obtain the integral. But I don't know how to show that the integral along this semicircle is 0 as R approaches infinity. There are two ways that I've seen my professor to do this is to use the inequality $\mid \int_{\gamma}f(z)dz \mid \leq sup \mid f(z) \mid length(\gamma)$. Sometimes, he uses the maximum likelihood estimate which I really have no idea about. Can someone help me on this? Are these two techniques always applicable and interchangable in this case and in some other similar integrals over some arc length. I really appreciate any help.
Try reading the last few posts on this.

3. Thanks a lot for that great tutorial on contour integration TPH. You have covered more details and more examples in this particular topic more than my textbook (Princeton lectures in analysis book 2). I see my professor uses the sup length inequality I mentioned above a lot of times, while your tutorial shows some clever ways to do the estimation depending on the function.

4. Originally Posted by namelessguy
Thanks a lot for that great tutorial on contour integration TPH. You have covered more details and more examples in this particular topic more than my textbook (Princeton lectures in analysis book 2). I see my professor uses the sup length inequality I mentioned above a lot of times, while your tutorial shows some clever ways to do the estimation depending on the function.
I plan on finishing it.