In (5.12.8) the fractional powers have their principal values when and , and are continued via continuity.
with the contour as shown in Figure 5.12.1.
when , is not an integer and the contour cuts the real axis between −1 and the origin. See Figure 5.12.2.
where the contour starts from an arbitrary point in the interval , circles 1 and then 0 in the positive sense, circles 1 and then 0 in the negative sense, and returns to . It can always be deformed into the contour shown in Figure 5.12.3.