The function (not ) has the following integral representations:
These representations are valid when , except (15.6.6) which holds for . In all cases the integrands are continuous functions of on the integration paths, except possibly at the endpoints. Note that (15.6.8) can be rewritten as a fractional integral. In addition:
In (15.6.1) all functions in the integrand assume their principal values.
In (15.6.2) the point lies outside the integration contour, and assume their principal values where the contour cuts the interval , and at .
In (15.6.3) the point lies outside the integration contour, the contour cuts the real axis between and , at which point and .
In (15.6.4) the point lies outside the integration contour, and at the point where the contour cuts the negative real axis and .
In (15.6.5) the integration contour starts and terminates at a point on the real axis between and . It encircles and once in the positive direction, and then once in the negative direction. See Figure 15.6.1. At the starting point and are zero. If desired, and as in Figure 5.12.3, the upper integration limit in (15.6.5) can be replaced by . However, this reverses the direction of the integration contour, and in consequence (15.6.5) would need to be multiplied by .
In (15.6.6) the integration contour separates the poles of and from those of , and has its principal value.
In (15.6.7) the integration contour separates the poles of and from those of and , and has its principal value.