§13.4 Integral Representations

For the function $J_{b-1}$ see §10.2(ii).

where $c$ is arbitrary, $\mathrm{\Re }c>0$. For the functions $K_{b-1}$ and ${}_2{}^{}𝐅_1^{}$ see §10.25(ii) and §§15.1, 15.2(i).

The contour of integration starts and terminates at a point $\alpha$ on the real axis between $0$ and $1$. It encircles $t=0$ and $t=1$ 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 $t=\alpha$. Similar conventions also apply to the remaining integrals in this subsection.

At the point where the contour crosses the interval $(1,\mathrm{\infty })$, $t^{-b}$ and the ${}_2{}^{}𝐅_1^{}$ 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 $\mathrm{ph}t=\pi$ and $\mathrm{ph}\left(1+t\right)=0$.

Again, $t^{-c}$ and the ${}_2{}^{}𝐅_1^{}$ function assume their principal values where the contour (see Figure 5.9.1) intersects the positive real axis.