- #1

disregardthat

Science Advisor

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Compute [tex]\int^{\infty}_0 \frac{x^{\alpha}}{1+x^2} dx[/tex] for some [tex]-1<\alpha<1[/tex].

EDIT: This was slightly wrong.

The hint given is that we can integrate from -p to p except for a small semi-circle around 0, and a large semicircle from p to -p, and choose a branch of [tex]z^{\alpha}[/tex]. Wouldn't this in any case be the method for computing [tex]\int^{\infty}_{-\infty} \frac{x^{\alpha}}{1+x^2} dx[/tex]?

I'd appreciate some help in choosing a proper contour. I'd know how to integrate from -infinity to infinity, but from 0 I have no idea. Also, can I choose my branch for the logarithm arbitrarily?

EDIT: This was slightly wrong.

The hint given is that we can integrate from -p to p except for a small semi-circle around 0, and a large semicircle from p to -p, and choose a branch of [tex]z^{\alpha}[/tex]. Wouldn't this in any case be the method for computing [tex]\int^{\infty}_{-\infty} \frac{x^{\alpha}}{1+x^2} dx[/tex]?

I'd appreciate some help in choosing a proper contour. I'd know how to integrate from -infinity to infinity, but from 0 I have no idea. Also, can I choose my branch for the logarithm arbitrarily?

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