# Contour integral w/ sqrt

1. Oct 10, 2011

1. The problem statement, all variables and given/known data
The integral I'm trying to solve is sqrt(x)/(1+x^2) from 0 to infinity.

2. Relevant equations

3. 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.

#### Attached Files:

• ###### contourintegral_sqrt.pdf
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2. Oct 10, 2011

### 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.

Last edited: Oct 10, 2011
3. Oct 10, 2011

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

4. Oct 10, 2011

### 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 $(1-e^{i 3 \pi/2}) = (1 + i)$.

5. Oct 10, 2011