On the half circunferemce (radius ) you mentioned, find:
surrounding the origin with a little circumference of radius in the upper plane. After, take limits .
Often when I solve a real integral in the complex plane I take a half circle in the upper plane an calculate the residues to get the integral. But what if theres a singular point in origo?
Here's an example of what I mean.
If anyone could show a method to solve this kind of integral I go.
Before I get my hands dirty. Even if it's not the primary issue, I rewrote the integral like this...
Now if I try to find the residues... How would I go about it. If I calculate the residue in origo in the upper plane, what about the lower plane..?
Should I just calculate all residues clockwise in the inner circle..? Something like this. where f(z) is given by the integral.
For 'semplicity's sake' as first step we can integrate the function 'counterclockwise' along the 'red path' of the figure...
... where the are the residues of inside ther red path. How many poles of there are inside the 'red path'?... of course the only pole is at so that is...
Now what is the next step?...
The point of this thread was to learn how to calculate integrals in the complex plane where theres a singularity in origo. If anyone would help to or show me how to do that I'd be thrilled.
Hi again! I was shown a how to solve exercises like this some weeks ago. Now I'm closing in on the exam and when I try to work through the solutions I get somewhat puzzled.
First step was to use Eulers formulas, in particular sin x.
Now he calculated the residues of each term by it self.
where f(z) i given by
where g(z) i given by
I know how to calculate the residues, but I don't understand why z = 0 and z = i goes to f(z). Or why z = 0 and z = -2i goes to g(z)..? Feels like I'm missing the main issue here... Any help would be appreciated!