Contour Integration with Square Root Function

[outhsakotad]

Homework Statement

The integral I'm trying to solve is sqrt(x)/(1+x^2) from 0 to infinity.

Homework Equations

The Attempt at a Solution

I've attached my solution. I know it's not right, as I shouldn't get an imaginary solution. The answer is actually pi/sqrt(2) according to my book. The author used a keyhole contour (branch cut along positive real axis and avoiding the branch point at z=0), but I don't see what's wrong with my approach. Any help would be greatly appreciated.

 

[Mute]

First consider: What is $i^{1/2}$ in cartesian form? (x + i*y). This won't solve the issue, but is just to remind you that to check that i^0.5 isn't actually 1- i (it's not, but it's close).

The real problem is the arc on the negative axis. You set $z = re^{i\pi}$, but forgot an extra factor of e^{i pi} from the differential dz.

 

[outhsakotad]

sqrt(i) = e^(i*pi/4)= (1/sqrt(2))*(1+i)... Any idea what is amiss? I still don't see it.

 

[Mute]

Whoops, you got in before I edited it. You missed a factor of e^{i pi} that comes from $dz = dr e^{i\pi}$ on your arc on the negative axis. You really have a factor of $(1-e^{i 3 \pi/2}) = (1 + i)$.

 

[outhsakotad]

Ah, okay. Thanks, so much. That solves some of the issues I was having with earlier problems too I think.