Some higher order Cauchy–Pompeiu integral representations

Abstract

Integral representation formulas play an essential role in the modern function theory. They serve to solve boundary value problems for differential equations. As an example of such integral representations is the Cauchy formula for analytic functions but it, thus, is just a special case of the Cauchy–Pompieu formula. Higher Cauchy–Pompieu formulas in the complex, hypercomplex and Clifford analysis have been presented by iteration in [Begehr, H., 2000, Integral representations for differentiable functions. Ricci, P. E. (Ed.), Current Problems in Analysis and Mathematical Physics. Papers of the 2nd international symposium dedicated to the memory of Prof. Gaetano Fichera (1922–1996). (Roma: Dipartimento di Matematica, Universita di Roma), 111–130, Begehr, H., 2002, Integral representations in complex, hypercomplex and Clifford analysis. Integral Transforms and Special Functions, 13(3), 223–241]. In this paper, certain special cases of higher order Cauchy–Pompeiu integral representations are given.