Improper Integral With Branch Cut

[Illuminerdi]

I understand most of the problem, but have yet to understand where a particular term came from. The problem is as follows:

Homework Statement

Show that (0 to ∞)∫dx/[(x²+1)√x] = π/√2
Hint: f(z)=z^−1/2/(z²+ 1) = e^{(−1/2) log z} /(z²+ 1).

The Attempt at a Solution

I actually have a solutions manual on me, but it's missing a step that I do not understand. I know to break the integral into 4 parts, an outer semi-circle contour that's infinitely large (of radius R), an inner contour that's infinitely small surrounding z=0 (of radius δ), a left contour from -R to δ, and a right contour from δ to R, but the solutions manual goes from parametrizing the left and right curves (by r) to combining them into a single integral with a factored term,
{δ to R} (1-i)∫dr/[(r²+1)√r].

I have no idea where this (1-i) term comes from.

 

[Mute]

Because you have a square root term in your expression there must be a branch cut in the complex plane in order to make the square root function single-valued. In order to choose the contour you have chosen, you have to choose the branch cut to run from z = 0 to infinity.

The value of the square root function is different above the branch and below it (there is an extra phase factor). Because of this, the lower segment of your contour running from R to delta does not cancel with the upper segment running from delta to R, but instead contributes a factor (1-i).

Have you studied branch points and branch cuts yet?