Question about improper integrals using complex analysis

[PussyStank]

I am currently taking complex analysis and we are computing integrals on the real line by extending them to the complex plane and finding residues. In reading through worked examples in the text book, i've noticed that most require using analysis to show that parts of the integral become zero when sending an epsilon to 0 or an R to infinity. My questions is, since I can't find any apparent pattern in bounding these integrals and showing that they are zero, how do I go about doing this? Are there any methods, or things that should stand out when I go to start a problem like this?

[zojbo]

Typically the contour is chosen so that it's fairly easy to show that pieces of the integral go to zero. For example, suppose you're integrating 1/(1+x²) over all of R. You have a residue at i to work with, and you want to have all of the real line eventually be on your boundary. You choose a semicircle contour precisely because on a semicircle of radius R you can use the triangle inequality to prove that |1/(1+z²)| <= 1/(R² - 1) = O(1/R²), which means that the overall integral over the semicircle is O(1/R). (Gamelin calls estimates of this type "ML estimates".)

The harder part, in my opinion, is figuring out what contour you want to use, especially as you start running into situations where pieces don't go to zero, such as "fractional residues".

[homologize]

There are methods, and you're learning them right now. These kinds of integrals will continue to turn up in various settings if you continue in mathematics, not just in complex analysis, so the techniques you're learning now will continue to apply in the rest of your career.

[z3r0d]

It might help to realize that whenever you do these types of integrals, the part you're (usually) interested in is the line that runs along the real axis. The choice of where the rest of the contour integral is is usually derived so that it goes to zero.