I've been going crazy with this complex integral I have to estimate with the Residues Theorem. I'm obviously missing a sign or something else, but I fear I may be commiting a conceptual mistake.
∫∞0dxlogxx2−1
I choose to solve this integral by computing the following complex integral:
∫γlog2zz2−1dz=limϵ→0[∫R0dxlog2(x+iϵ)(x+iϵ)2−1+∫0Rdxlog2(x−iϵ)(x−iϵ)2−1+
+∫02πdθiϵeiθlog2(ϵeiθ)(ϵeiθ)2−1+∫02πdθiϵeiθlog2(1+ϵeiθ)(1+ϵeiθ)2−1]+∫˜γdzlog2zz2−1
Where γ is the "keyhole" path, and ˜γ is the circle centered in z=0 with R radius. If R→∞, then:
=∫∞0dzlog2(x)x2−1−∫∞0dx(log(x)+2πi)2x2−1
=−4πi∫∞0dxlog(x)x2−1+4π2∫∞0dx1x2−1=−4πi∫∞0dxlog(x)x2−1
since ∫∞0dx1x2−1=0.
I can estimate the complex integral with the Residues theorem:
∫γlog2zz2−1dz=2πiRes[f(z),−1]=iπ3
and this means that:
∫∞0dxlog(x)x2−1=−π24
which is wrong, since the correct result should be π24. Is there something wrong with my procedure?
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