General Integral Representations of Parabolic Functions and of their Products

Abstract

Such integral representations will henceforth be understood as integrals having the form of Eq. (2.12). Our immediate objective is to also determine this type of integral for the functions W, _{xμ /2}(z) and W _{x,μ /2}(z. e ^±πi). To this end we make use of the s-form of the integral, mentioned above for the function M _{x ,μ/2}(z) having the convergence condition Re((1 + μ)/2 ± x) > 0, and deform its path of integration by moving a central point of this path to an infinitely remote position along a line inclined at an angle σ, |σ| < π, with respect to the real axis. In Fig. 3 we show an intermediate stage in this path deformation procedure in addition to the final path of integration. Also shown are the phase angles of the points which are used in the altered integral. According to the sign convention of arc(s ± 1) thus adopted, it is clear that 1 + s = s + 1 and 1 − s = (s − 1) e^−πi has to be substituted in the original integral.