# §13.4 Integral Representations

## §13.4(i) Integrals Along the Real Line

For the function see §10.2(ii).

where is arbitrary, . For the functions and see §10.25(ii) and §§15.1, 15.2(i).

## §13.4(ii) Contour Integrals

Figure 13.4.1: Contour of integration in (13.4.11). (Compare Figure 5.12.3.)

The contour of integration starts and terminates at a point on the real axis between 0 and 1. It encircles and once in the positive sense, and then once in the negative sense. See Figure 13.4.1. The fractional powers are continuous and assume their principal values at . Similar conventions also apply to the remaining integrals in this subsection.

13.4.12, .

At the point where the contour crosses the interval , and the function assume their principal values; compare §§15.1 and 15.2(i). A special case is

The contour cuts the real axis between −1 and 0. At this point the fractional powers are determined by and .

13.4.15.

Again, and the function assume their principal values where the contour intersects the positive real axis.

## §13.4(iii) Mellin–Barnes Integrals

If , then

where the contour of integration separates the poles of from those of .

If and , then

where the contour of integration separates the poles of from those of .

where the contour of integration passes all the poles of on the right-hand side.