A Connection Between Elementary Functions and Higher Transcendental Functions

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 ₂ F₁ (a,b;c;x) is an average of z- a over the line segment joining $1-x$ and 1, the confluent hypergeometric function ₁ F₁ (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 ₂ F₁ and ₁ F₁ 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.