A Cauchy integral formula is constructed for solutions to the polynomial Dirac equation (Dk+Yfcrn~JQ bmDm)f = 0 , where each bm is a complex number, D is the Dirac operator in R" , and f is defined on a domain in R" and takes values in a complex Clifford algebra. Some basic properties for the solutions to this equation, arising from the integral formula, are described, including an approximatio...

We develop two types of integral formulas for the perimeter of a convex body K in planar geometries. We derive Cauchy-type formulas for perimeter in planar Hilbert geometries. Specializing to H2 we get a formula that appears to be new. In the projective model of H2 we have P = (1/2) ∫ w dφ. Here w is the Euclidean length of the projection of K from the ideal boundary point R = (cosφ, sinφ) onto...

We obtain a flagged form of the Cauchy determinant and establish a correspondence between this determinant and nonintersecting lattice paths, from which it follows that Cauchy identity on Schur functions. By choosing different origins and destinations for the lattice paths, we are led to an identity of Gessel on the Cauchy sum of Schur functions in terms of the complete symmetric functions in t...

Euclidean Clifford analysis is a higher dimensional function theory offering a refinement of classical harmonic analysis. The theory is centred around the concept of monogenic functions, which constitute the kernel of a first order vector valued, rotation invariant, differential operator ∂ called the Dirac operator, which factorizes the Laplacian. More recently, Hermitean Clifford analysis has ...

Here we will consider examples of conformally flat manifolds that are conformally equivalent to open subsets of the sphere S. For such manifolds we shall introduce a Cauchy kernel and Cauchy integral formula for sections taking values in a spinor bundle and annihilated by a Dirac operator, or generalized Cauchy-Riemann operator. Basic properties of this kernel are examined, in particular we exa...

We develop the basic elements of a Cauchy theory on the complex Levi-Civita field, which constitutes the smallest algebraically closed nonArchimedean extension of the complex numbers. We introduce a concept of analyticity based on differentiation, and show that it leads to local expandability in power series. We show that analytic functions can be integrated over suitable piecewise smooths path...