Calculate the following integral using Residue Theorem
from to
Hello,
Let
Obviously, i and -i are double poles, because
So, using the general formula for a pole of order n :
we easily get the residue of f at i (the one that will be useful for the computation) :
which is easy to compute.
Spoiler:
Now, use the semicircle contour :
And keep the poles with positive imaginary part. So only i.
And then
Hey...
So
You got wrong for the second one too.
Isn't it surprising to get a negative answer for the integral of a positive function ?
Yes. If we had the integral from infinity to -infinity, it would have been more logic to take the semi circle and the negative side of the imaginary.Am I right to say that you took semi contour and the positive side of the imaginary because of the -infinity to infinity?
But the result is actually the same since
But more commonly, we take the upper semi circle
I hope I'm clear :s