Repeat exercise 20 this time with \(I_1 = \int_C \frac{z^p}{z^2+1}\,\mathrm{d}z\) and \(I_2 = \int_0^\infty \frac{x^p}{x^2+1}\,\mathrm{d}x\) with \(0 < p < 1\); the contour is the same as before.
Exercise 20 of the source reads: (a) 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. (b) 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 \(I_2\).
Figure 3 of the source shows the contour \(C = L_+ \cup C_R \cup L_- \cup \gamma_\epsilon\): \(L_+\) runs just above the positive real axis from the small circle \(\gamma_\epsilon\) around \(0\) out to the large circle \(C_R\) of radius \(R\) centered at \(0\), \(C_R\) is traversed counterclockwise, \(L_-\) returns just below the positive real axis, and \(\gamma_\epsilon\) is traversed clockwise around \(0\).