- #1
Dixanadu
- 254
- 2
Homework Statement
Hey everyone,
So here's the problem, nice and simple. I have to find the following integral:
[itex]\int_{0}^{\infty} \frac{x^{p}ln(x)}{x^{2}+1}dx, 0<p<1[/itex]
Homework Equations
The only thing relevant is the residue theorem:
[itex]\oint_{c}f(z)=2\pi i \times[/itex] sum of residues enclosed
The Attempt at a Solution
So I've gone through it using a branch cut so that [itex]0<\theta<2\pi[/itex]. I then replaced x with [itex]z=re^{i(\theta + 2in\pi)}[/itex], setting n = 0 when integrating from 0 to infinity for Im(z)>0, then setting n = 1 when integrating from infinity back to 0 when Im(z)<0. Then I get this:
[itex]I=-\pi^{2}/2 \frac{cos(\pi p/2)}{sin(\pi p)e^{i\pi p}}[/itex]
Of course I can simplify it but its definitely wrong cos its negative, and there is a dependence on a complex exponential...so what have I done wrong? Am I even supposed to be using a branch cut or is it enough to do this with a contour indented at the singularity z = i in the upper half of the complex plane...? I don't see how that would work though, I am pretty sure branch cut is the way to go, but I get this...