Saturday, 25 February 2017

integration - Complex integral with Residues Theorem

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.



0dxlogxx21



I choose to solve this integral by computing the following complex integral:



γlog2zz21dz=limϵ0[R0dxlog2(x+iϵ)(x+iϵ)21+0Rdxlog2(xiϵ)(xiϵ)21+
+02πdθiϵeiθlog2(ϵeiθ)(ϵeiθ)21+02πdθiϵeiθlog2(1+ϵeiθ)(1+ϵeiθ)21]+˜γdzlog2zz21




Where γ is the "keyhole" path, and ˜γ is the circle centered in z=0 with R radius. If R, then:



=0dzlog2(x)x210dx(log(x)+2πi)2x21
=4πi0dxlog(x)x21+4π20dx1x21=4πi0dxlog(x)x21



since 0dx1x21=0.



I can estimate the complex integral with the Residues theorem:



γlog2zz21dz=2πiRes[f(z),1]=iπ3




and this means that:



0dxlog(x)x21=π24



which is wrong, since the correct result should be π24. Is there something wrong with my procedure?

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