How to choose contour in $\mathbb{C}$ to do Residue Integration.

Question

I'm almost sure that there's not any simple way to answer this question, but I'll try. I'm studying complex variables and the method of calculating improper integrals with residues but I'm struggling a little with the contours.

The teacher said that there is no general way to choose the contour, we can be lucky to find it and be able to solve the integral, or we don't solve the integral. And this sounded to me like finding $\delta$'s in proofs of continuity. This is a good example of what I'm trying to find out.

Although there's no "formula" to find $\delta$ for every function, there are some steps, that when we take we can figure out our $\delta$ in one almost easy way in many cases. What we do is: we bound $|f(x)-f(a)|$ and try to make appear there things like $|x-a|$, $|x+a|$ and $|x|$, because all of those we can bound.

After that, we look again, if there are things like $|x|$ and $|x+a|$, we enforce $|x-a|<1$ or something else, just to bound those things again. After that we look again, and we try to figure out what $|x-a|$ should be less than in order to make the whole thing less than $\epsilon$.

Now, when I've learned about continuity in the beginning of the course the teacher said the same: there's no way, or you have creativity to choose $\delta$ or forget about it. But these are guidelines that helped me in the vast majority of cases, even in $\Bbb C$ and $\Bbb R^n$.

So, is there a general method, general guidelines that I can follow to see what contour I should use? Since there can be lots of things to say about it, a reference explaining how to think about this and giving general techniques would be of great help.

Thanks very much in advance.

Answer 1

Your continuity guideline is very correct. Now,if i understood correctly, you want to find a contour $C$ in order to integrate a function $f$ on it using the Residues Theorem. Do you have anything more specific,an example for example:P.

For instance, let an $a\in \Bbb C$($a$ is not an isolated anomaly of $f$) and find the isolated anomalies of $f$(isolated poles) and find the one that is closer to $a$. Then you can integrate $f$ on the contour of the disk with centre $a$ and with radius being the distance of the closest anomaly of $f$ to $a$. Then $f$ can be integrated withour any trouble because there are not anomalies inside the disk.

The same you can do if you want to have a pole of $f$ inside your disk. Just make larger the radius in order to get some poles.

Also have in mind the independence of the paths during the integrate.

Don't know if i 've helped you.