$\int_0^\infty \frac{dx}{(x+i)\sqrt x}$ by a keyhole contour

Compute the integral \(I _ { 1 } = \int _ { C } \frac { \mathrm { d } z } { ( z + i ) \sqrt { z } }\) , where the contour C is shown in figure 3, \(C = L _ { + } \cup C _ { R } \cup L _ { - } \cup \gamma _ { \epsilon }\) , and we send the radius of \(\gamma _ { \epsilon }\) to zero and the radius of \(C _ { R }\) to infinity. Note that, because of the square root, \(z = 0\) is a branch point. We choose to have a branch cut on the positive real axis.

Find a relation between the integral \(I _ { 1 }\) and \(I _ { 2 } = \int _ { 0 } ^ { \infty } \frac { 1 } { ( x + i ) \sqrt { x } } \mathrm { d } x\) ,then use the result from the previous point to find the value of \(I _ { 2 }\) .