Hi,
while trying to evaluate below
I was hoping to use
1. Use complex numbers i.e pole at x=0
which gives 0
2. Expand by sin(x) by Taylor series around 0 and multiply by x
this gives a divergent series
which one is correct?
thanks
neither.
here is how you do it. note that
Because of the pole of the new integrand at , we instead compute
now continue with the integration using a semi-circular contour in the upper half plane, based on the real axis.
with sum work, i suppose your second method could work. but you multiply by 1/x not x
really? and what method is that? it seems like you were trying to apply the residue theorem, but it looks strange to me. are you sure you computed the residues correctly?
on the contrary, the second integrand has no poles.2. As I understand
and
both have poles at x = 0. Why did u take latter? Would u pls elaborate?
yes, I was referring to the Residue theorem.really? and what method is that? it seems like you were trying to apply the residue theorem, but it looks strange to me. are you sure you computed the residues correctly?
Res(sin(x)/x) at x=0 is 0 isn't it?
and
both have poles at x=0. Could u pls explain with some basic principles why it is not?
thanks
yes, it is 0. but the conditions to apply the residue theorem to this function directly are not fulfilled. that's why we needed to tweak things.
the second has a removable singularity at x = 0, not a pole. look up what it means to have a pole and a removable singularity. you will see that the two integrals are different in that respect.
and
both have poles at x=0. Could u pls explain with some basic principles why it is not?
thanks
Thank you very much for pointing me in the correct direction.
To apply the Residue theorem can't we create a contour like the one in the image?
File:Contour of KKR.svg - Wikipedia, the free encyclopedia