‎Complex-valued harmonic functions that are univalent and‎ ‎sense-preserving in the open unit disk $U$ can be written as form‎ ‎$f =h+bar{g}$‎, ‎where $h$ and $g$ are analytic in $U$‎. ‎In this paper‎, ‎we introduce the class $S_H^1(beta)$‎, ‎where $1<betaleq 2$‎, ‎and‎ ‎consisting of harmonic univalent function $f = h+bar{g}$‎, ‎where $h$ and $g$ are in the form‎ ‎$h(z) = z+sumlimits_{n=2}^inf...

The notion of a conjugate harmonic pair in the context of Hermitean Clifford analysis is introduced as a pair of specific harmonic functions summing up to a Hermitean monogenic function in an open region Ω of C. Under certain geometric conditions on Ω the conjugate harmonic to a given specific harmonic is explicitly constructed and the potential or primitive of a Hermitean monogenic function is...

We propose here a new approach for finding harmonic functions of killed random walks with small steps in the quarter plane. It is based on solving a certain functional equation that satisfies the generating function of the values taken by the harmonic functions. As a first application of our results, we obtain a simple expression for the harmonic function that governs the asymptotic tail distri...

We show that we can approximate every function f ∈ C(B1) with a s-harmonic function in B1 that vanishes outside a compact set. That is, s-harmonic functions are dense in C loc. This result is clearly in contrast with the rigidity of harmonic functions in the classical case and can be viewed as a purely nonlocal feature.

The general equation from previous work is specialized to a quadratic potential V (r) = −a +