A Connection Between Elementary Functions and Higher Transcendental Functions - Carlson, B. C. (SIAM journal on applied mathematics, 1969)
If f is holomorphic on a domain D in the complex plane, an analogous function F of several complex variables is constructed by taking a weighted average of f over the convex hull of {z1, z2, ⋯, zk}. Although F is defined at first only if the convex hull is contained in D, it is shown later that F can be continued analytically along any rectifiable arc in Dk, provided that singular points with zi = zj (for some distinct i, j) are excluded if D is multiply connected. Taylor and Laurent series for f have single-series analogues for F, and the analogue of Cauchy's integral formula is a representation of F by an integral around a contour in D encircling z1, z2, ⋯, zk. The hypergeometric function 2F1(a, b; c; x) is an average of z-a over the line segment joining 1 - x and 1, the confluent hypergeometric function 1F1(b; c; x) is an average of ez over the line segment joining x and 0, and elliptic integrals are averages of a half-odd-integral power of z over a triangle (or a quadrilateral for integrals of the third kind). The average of z-a in the case of k complex variables is the hypergeometric R-function, which appears in both the series and contour-integral representations of F. The parameters b and c in the 2F1 and 1F1 functions come from the weight function used in the averaging process. Even in the case of k variables the weight function is taken to have a rather special form, with the result that F always satisfies a system of Euler-Poisson partial differential equations. Connections with axially symmetric potential theory, fractional integration, and integral transforms are mentioned briefly.