0

I am trying to solve the following integral using a contour (large semi-circle connected to smaller semi-circle in the upper-half plane):

I have split the contour into 4 parts - the large semi-circle, the small semi-circle, the part on the negative real axis and the part on the positive real axis.

But something goes wrong with me and I can't come up with the result, which is the correct answer is $5\pi^5/32$.

So far what I have is the following:

∮f(z)dz=K1+K2+K3+K4 See image to notice who it is ∮f(z)dz https://i.stack.imgur.com/PuuN7.png

The point is, I'm at a loss to solve k2 and k4. It may be something simple but I still don't see the way.

CentraM
  • 1
  • 1
  • 2
    Well, show us the computations you did, then it will be easier to say where you are wrong. Or at least write the part you are unsure about. – Mark Mar 21 '24 at 20:52
  • 2
    https://math.stackexchange.com/questions/3007258/contour-integration-complex-analysis-int-0-infty-frac-log4x1x2?rq=1 – Benjamin Dickman Mar 21 '24 at 21:03
  • 3
    Integrals of the form $$\int^\infty_0\log^nxQ(x)\mathrm d x,$$ where $Q$ is a rational function having no positive/ negative pole may be computed by considering the integral $$\int_C\log^{n+1}xQ(x)\mathrm d z$$ with the keyhole contour $C$. – Zhang Mar 21 '24 at 21:17

0 Answers0