OK so I am not sure if this problem is analysis, number theory or calculus.
I am reading an English translation of Riemann's 1859 paper on the number of primes and cannot follow this bit.
He says consider this integral:
From infinity in a positive sense around a domain that includes 0 but does not include any other point of discontinuity. This is then equal to:
I would like to fill in the details of this step but have been unsuccessful. I have tried:
Finding the residue at z=0 by solving:
I also tried generalising x to z and integrating around the path from inf, to to to to
where
This gave interesting results when either; tended to 0 equals pi or equals pi/2.
But non-the less I have not been able to get the required result. Is one of my approaches right? Is there a better approach?
It's hard to see what is going on here without knowing more of the context, but the expression suggests to me that it means the integral of around a contour that goes from to along the real axis, then goes clockwise all the way round a circle of radius , and finally back along the real axis from to .
Notice that the two integrals going in opposite directions along the real axis will not cancel each other out, because , and will move onto another branch when z goes round the origin.
The contour is allowed to be any closed contour starting at x=inf that does not enclose any singularity except for the one at x=0. I only described a contour using to demonstrate that I have had a fair crack at solving this myself!
The exact wording from the paper is:
If one now considers the integral
from +inf to +inf taken in a positive sense around a domain which includes the value 0 but no other point of discontinuity of the integrand in its interior, then ths is easily seen to be equal to
provided that, in the many-valued function , the logarithm is determined so as to be real when x is negative.
Absolutely. Replace the hankel contour with it's mirror image and by doing so, elimiate the awkward in the numerator and simultaneously convert the integration to the more "normal" principal branch-cut integral along the negative real axis. When you do this, you obtain the integral:
Now make the usual substitutions for a branch-cut integral over the principle branch-cut of log(z) along the negative real axis: starting along the bottom trace at , let , around the origin, let and along the top trace, let . Also, don't forget, you need to analyze this under the assumption that , then show that as the integral around the origin goes to zero.