This paper recalls the work of D. Pompeiu who introduced the notion of set distance in his thesis published one century ago. The notion was further studied by F. Hausdorff, C. Kuratowski who acknowledged in their books the contribution of Pompeiu and it is frequently called the Hausdorff distance. keywords: Hausdorff distance, Hausdorff-Pompeiu distance, Pompeiu functions, Pompeiu conjecture, S...

In this paper, we combine the fractional ψ − $$ \psi - hyperholomorphic function theory with calculus respect to another function. As a main result, Borel–Pompeiu type formula related Fueter operator vector-valued is proved.

We introduce a family of Cauchy integral formulas for slice and slice regular functions on a real associative *-algebra. For every suitable choice of a real subspace of the algebra, a different formula is given, in which the domains of integration are subsets of the subspace. In particular, in the quaternionic case we get a volume Cauchy formula. In the Clifford algebra case, the choice of the ...

In this paper we show how to construct a regular, non commutative Cauchy kernel for slice regular quaternionic functions. We prove an (algebraic) representation formula for such functions, which leads to a new Cauchy formula. We find the expression of the derivatives of a regular function in terms of the powers of the Cauchy kernel, and we present several other consequent results. AMS Classific...

We introduce the Cauchy augmentation operator for basic hypergeometric series. Heine’s 2φ1 transformation formula and Sears’ 3φ2 transformation formula can be easily obtained by the symmetric property of some parameters in operator identities. The Cauchy operator involves two parameters, and it can be considered as a generalization of the operator T (bDq). Using this operator, we obtain extensi...

The construction of analogues of the Cauchy kernel is crucial for the solution of Riemann–Hilbert problems on compact Riemann surfaces. A formula for the Cauchy kernel can be given as an infinite sum over the elements of a Schottky group, and this sum is often used for the explicit evaluation of the kernel. In this paper a new formula for a quasi-automorphic analogue of the Cauchy kernel in ter...

In this paper, we mainly study the integral representations for functions f with values in a universal Clifford algebra C(Vn,n), where f ∈ Λ(f,Ω), Λ(f,Ω) = { f |f ∈ C∞(Ω, C(Vn,n)),max x∈Ω ∣∣Djf(x)∣∣ = = O(M )(j → +∞), for someM, 0 < M < +∞} . The integral representations of Tif are also given. Some properties of Tif and Πf are shown. As applications of the higher order Pompeiu formula, we get t...

We demonstrate connections that exists between a conjecture of Schiffer’s (which is equivalent to a positive answer to the Pompeiu problem), stationary solutions to the Euler equations, and the convergence of solutions to the Navier-Stokes equations to that of the Euler equations in the limit as viscosity vanishes. We say that a domain Ω ⊆ R, d ≥ 2, has the Pompeiu property if, given that the i...