A continuous function f u iv is a complex-valued harmonic function in a simply connected complex domain D ⊂ C if both u and v are real harmonic in D. It was shown by Clunie and Sheil-Small 1 that such harmonic function can be represented by f h g, where h and g are analytic in D. Also, a necessary and sufficient condition for f to be locally univalent and sense preserving in D is that |h′ z | >...

Let f be a function on a bounded domain Ω ⊆ R and δ be a positive function on Ω such that B(x, δ(x)) ⊆ Ω. Let σ(f)(x) be the average of f over the ball B(x, δ(x)). The restricted mean-value theorems discuss the conditions on f, δ, and Ω under which σ(f) = f implies that f is harmonic. In this paper, we study the stability of harmonic functions with respect to the map σ. One expects that, in gen...

and Applied Analysis 3 2. Preliminaries A continuous function f u iv is a complex-valued harmonic function in a domain D ⊂ if both u and v are real harmonic in D. In any simply connected domain, we can write

The cluster assumption had a significant impact on the reasoning behind methods of semi-supervised classification in graph-based learning. The literature includes numerous applications where harmonic functions provided estimates that conformed to data satisfying this well-known assumption, but the relationship between this assumption and harmonic functions is not as well-understood theoreticall...

‎By observing the equivalence of assertions on determining the jump of a‎ ‎function by its differentiated or integrated Fourier series‎, ‎we generalize a‎ ‎previous result of Kvernadze‎, ‎Hagstrom and Shapiro to the whole class of‎ ‎functions of harmonic bounded variation‎. ‎This is achieved without the finiteness assumption on‎ ‎the number of discontinuities‎. ‎Two results on determination of ...

The many-body wave function of a system of interacting particles confined by a time-dependent harmonic potential and perturbed by a time-dependent spatially homogeneous electric field is derived via the Feynman path-integral method. The wave function is comprised of a phase factor times the solution to the unperturbed time-dependent Schrödinger equation with the latter being translated by a tim...

The harmonic function F exp(j2rvt) plays an important role in science and engineering. It has frequency v and complex amplitude F. Its real part IFIcos(2~vt + arg{F}) is a cosine function with amplitude jF( and phase arg{F}. The variable t usually represents time; the frequency v has units of cycles/s or Hz. The harmonic function is regarded as a building block from which other functions may be...

A new approach for frequency analysis of recorded signals and readout the frequency of harmonics is presented in the paper. The main purpose has been achieved by the cross-correlation function and Hilbert transform. Using the method presented in the paper, there is another possibility to observe and finally to identify single harmonic apart from commonly used Fourier transform. Identification o...

We shall give a lower estimate of harmonic‎ ‎functions of order greater than one in a half space‎, ‎which‎ ‎generalize the result obtained by B‎. ‎Ya‎. ‎Levin in a half plane‎.

Let p be a real number greater than one and let G be a connected graph of bounded degree. In this paper we introduce the p-harmonic boundary of G. We use this boundary to characterize the graphs G for which the constant functions are the only p-harmonic functions on G. It is shown that any continuous function on the p-harmonic boundary of G can be extended to a function that is p-harmonic on G....