Related papers: Noncommutative Cauchy integral formula
This study is on Cauchy's function $f(z)$ and its integral, $J[f(z)]\equiv (2\pi i)^{-1}\oint_C f(t)dt/(t-z)$ taken along a closed simple contour $C$, in regard to their comprehensive properties over the entire $z=x+iy$ plane consisted of…
The slice Dirac operator over octonions is a slice counterpart of the Dirac operator over quaternions. It involves a new theory of stem functions, which is the extension from the commutative $ O(1) $ case to the non-commutative $ O(3) $…
We prove a Cauchy identity for free quasi-symmetric functions and apply it to the study of various bases. A free Weyl formula and a generalization of the splitting formula are also discussed.
We give a local characterization of the class of functions having positive distributional derivative with respect to $\bar{z}$ that are almost everywhere equal to one of finitely many analytic functions and satisfy some mild non-degeneracy…
The Bloch-Landau Theorem is one of the basic results in the geometric theory of holomorphic functions. It establishes that the image of the open unit disc $\mathbb{D}$ under a holomorphic function $f$ (such that $f(0)=0$ and $f'(0)=1$)…
In this paper, we study integral functionals defined on spaces of functions with values on general (non-separable) Banach spaces. We introduce a new class of integrands and multifunctions for which we obtain measurable selection results.…
A crucial extension of quaternionic function theory to octonions is the concept of slice regular functions, introduced to handle holomorphic-like properties in a non-associative setting. In this paper, first we present a generalization of…
In this paper we propose an Almansi-type decomposition for slice regular functions of several quaternionic variables. Our method yields $2^n$ distinct and unique decompositions for any slice function with domain in $\mathbb{H}^n$. Depending…
We present a number of results relating partial Cauchy-Littlewood sums, integrals over the compact classical groups, and increasing subsequences of permutations. These include: integral formulae for the distribution of the longest…
In the article the class of slice regular functions is shown to be closed under a new regular composition. The new regular composition turns out to be globally defined in contrast to the locally defined version by Vlacci. Its advantage over…
This paper introduces noncommutative analogs of monomial symmetric functions and fundamental noncommutative symmetric functions. The expansion of ribbon Schur functions in both of these basis is nonnegative. With these functions at hand,…
We describe partial differential operators for which we can construct generalised integral means satisfying Pizzetti-type formulas. Using these formulas we give a new characterisation of summability of formal power series solutions to some…
This paper is the continuation of the paper arXiv:1509.06950, which is Part I under the same title. In this paper, we prove a generalized Cauchy formula for the integrals of logarithmic forms on products of projective lines, and give an…
The aim of this paper is to prove that a large class of quaternionic slice regular functions result to be (ramified) covering maps. By means of the topological implications of this fact and by providing further topological structures, we…
In this paper we introduce the notion of $\mathcal I^{\mathcal K}$-Cauchy function, where $\mathcal I$ and $\mathcal K$ are ideals on the same set. The $\mathcal I^{\mathcal K}$-Cauchy functions are a generalization of $\mathcal I^*$-Cauchy…
We extend some definitions and give new results about the theory of slice analysis in several quaternionic variables. The sets of slice functions which are respectively slice, slice regular and circular w.r.t. given variables are…
Very recently, the concept of generalized partial-slice monogenic (or regular) functions has been introduced to unify the theory of monogenic functions and of slice monogenic functions over Clifford algebras. Inspired by the work of A.…
We survey recent work and announce new results concerning two singular integral operators whose kernels are holomorphic functions of the output variable, specifically the Cauchy-Leray integral and the Cauchy-Szeg\H o projection associated…
Formul{\ae} are derived for expressing Cauchy and Hilbert transforms of a function $f$ in terms of Cauchy and Hilbert transforms of $f(x^r)$. When $r$ is an integer, this corresponds to evaluating the Cauchy transform of $f(x^r)$ at all…
In this paper we derive a Cauchy integral formula for holomorphic and hyperholomorphic functions in domains bounded by a Koch snowflake in two and three dimensional setting.