ABSTRACT
Fox’s H function is a general function defined by the Mellin–Barnes integral (12.1). It was introduced by Fox [4] in the context of symmetrical Fourier kernels and has been studied by a number of authors thereafter [1,9,11,14]. Fox’s H function has a number of important applications, most notably in statistics [2,5,11] and fractional calculus [10,11]. Braaksma’s mammoth manuscript [1] remains among the deepest investigations on the topic. In this note we will be only interested in the case when the parameter μ defined in (12.2) equals zero (Braaksma’s work contains a careful study of both cases μ > 0 and μ = 0, while the case μ < 0 reduces to μ > 0 by a simple change of variable). When μ = 0 the integral (12.1) defining the H function only converges if |z| > β −1, where β is given in (12.2). It also converges for |z| > β −1 but over a different integration contour so that the two functions obtained in this way are not, generally speaking, analytic continuations of each other. Braaksma constructed analytic continuations of the H function to an infinite set of sectors forming a partition of the Riemann surface of the logarithm. The points of intersection of the circle |z| = β −1 with the rays bounding these sectors are singular points of H. He further derived analytic continuation from each such sector to the domain |z| > β −1 on this Riemann surface.
