scholarly journals The Harmonicity of Slice Regular Functions

2020 ◽  
Author(s):  
Cinzia Bisi ◽  
Jörg Winkelmann
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Regular Function ◽  
Representation Formula ◽  
Mean Value ◽  
Quaternionic Analysis ◽  
Poisson Formula ◽  
Slice Regular Functions ◽  
Regular Functions ◽  
Mean Value Formula

Abstract In this article, we investigate harmonicity, Laplacians, mean value theorems, and related topics in the context of quaternionic analysis. We observe that a Mean Value Formula for slice regular functions holds true and it is a consequence of the well-known Representation Formula for slice regular functions over $${\mathbb {H}}$$ H . Motivated by this observation, we have constructed three order-two differential operators in the kernel of which slice regular functions are, answering positively to the question: is a slice regular function over $${\mathbb {H}}$$ H (analogous to an holomorphic function over $${\mathbb {C}}$$ C ) ”harmonic” in some sense, i.e., is it in the kernel of some order-two differential operator over $${\mathbb {H}}$$ H ? Finally, some applications are deduced such as a Poisson Formula for slice regular functions over $${\mathbb {H}}$$ H and a Jensen’s Formula for semi-regular ones.

scholarly journals Almansi Theorem and Mean Value Formula for Quaternionic Slice-regular Functions

2020 ◽  
Vol 30 (4) ◽  
Author(s):  
Alessandro Perotti
Keyword(s):  
Harmonic Functions ◽  
Mean Value ◽  
Zonal Harmonic ◽  
Slice Regular Functions ◽  
Regular Functions ◽  
Quaternionic Polynomials ◽  
Mean Value Formula

Abstract We prove an Almansi Theorem for quaternionic polynomials and extend it to quaternionic slice-regular functions. We associate to every such function f, a pair $$h_1$$ h 1 , $$h_2$$ h 2 of zonal harmonic functions such that $$f=h_1-\bar{x} h_2$$ f = h 1 - x ¯ h 2 . We apply this result to get mean value formulas and Poisson formulas for slice-regular quaternionic functions.

scholarly journals A New Approach to Slice Analysis Via Slice Topology

2021 ◽  
Vol 31 (5) ◽  
Author(s):  
Xinyuan Dou ◽  
Ming Jin ◽  
Guangbin Ren ◽  
Irene Sabadini
Keyword(s):  
Taylor Expansion ◽  
Regular Function ◽  
Alternative Algebra ◽  
Representation Formula ◽  
Monogenic Functions ◽  
Slice Regular Functions ◽  
New Approach ◽  
Regular Functions ◽  
Several Variables ◽  
Slice Monogenic Functions

AbstractIn this paper we summarize some known facts on slice topology in the quaternionic case, and we deepen some of them by proving new results and discussing some examples. We then show, following Dou et al. (A representation formula for slice regular functions over slice-cones in several variables, arXiv:2011.13770, 2020), how this setting allows us to generalize slice analysis to the general case of functions with values in a real left alternative algebra, which includes the case of slice monogenic functions with values in Clifford algebra. Moreover, we further extend slice analysis, in one and several variables, to functions with values in a Euclidean space of even dimension. In this framework, we study the domains of slice regularity, we prove some extension properties and the validity of a Taylor expansion for a slice regular function.

scholarly journals Spherical Coefficients of Slice Regular Functions

2021 ◽  
Vol 76 (4) ◽  
Author(s):  
Amedeo Altavilla
Keyword(s):  
Differential Equations ◽  
Regular Function ◽  
Countable Family ◽  
Slice Regular Functions ◽  
Regular Functions ◽  
Spherical Expansion ◽  
Derivatives Of

AbstractGiven a quaternionic slice regular function f, we give a direct and effective way to compute the coefficients of its spherical expansion at any point. Such coefficients are obtained in terms of spherical and slice derivatives of the function itself. Afterwards, we compare the coefficients of f with those of its slice derivative $$\partial _{c}f$$ ∂ c f obtaining a countable family of differential equations satisfied by any slice regular function. The results are proved in all details and are accompanied to several examples. For some of the results, we also give alternative proofs.

scholarly journals On the real differential of a slice regular function

2018 ◽  
Vol 18 (1) ◽  
pp. 5-26 ◽  
Author(s):  
Amedeo Altavilla
Keyword(s):  
General Result ◽  
Regular Function ◽  
Singular Set ◽  
Slice Regular Functions ◽  
The Real ◽  
Regular Functions ◽  
New Information ◽  
Spherical Expansion ◽  
New Formula ◽  
Derivatives Of

AbstractIn this paper we show that the real differential of any injective slice regular function is everywhere invertible. The result is a generalization of a theorem proved by G. Gentili, S. Salamon and C. Stoppato and it is obtained thanks, in particular, to some new information regarding the first coefficients of a certain polynomial expansion for slice regular functions (calledspherical expansion), and to a new general result which says that the slice derivative of any injective slice regular function is different from zero. A useful tool proven in this paper is a new formula that relates slice and spherical derivatives of a slice regular function. Given a slice regular function, part of its singular set is described as the union of surfaces on which it results to be constant.

A class of symmetric ordinary differential operators whose deficiency numbers differ by an integer†

1978 ◽  
Vol 82 (1-2) ◽  
pp. 117-134 ◽  
Author(s):  
Richard C. Gilbert
Keyword(s):  
Differential Operator ◽  
Positive Integer ◽  
Asymptotic Expansions ◽  
Differential Operators ◽  
Ordinary Differential Operator ◽  
Ordinary Differential Operators ◽  
Half Axis ◽  
Complex Coefficients

SynopsisFormulas are determined for the deficiency numbers of a formally symmetric ordinary differential operator with complex coefficients which have asymptotic expansions of a prescribed type on a half-axis. An implication of these formulas is that for any given positive integer there exists a formally symmetric ordinary differential operator whose deficiency numbers differ by that positive integer.

scholarly journals DIFFUSIVE REPRESENTATIONS FOR THE ANALYSIS AND SIMULATION OF FLARED ACOUSTIC PIPES WITH VISCO-THERMAL LOSSES

2006 ◽  
Vol 16 (04) ◽  
pp. 503-536 ◽  
Author(s):  
TH. HÉLIE ◽  
D. MATIGNON
Keyword(s):  
Finite Order ◽  
Acoustic Waves ◽  
Complex Analysis ◽  
Differential Operators ◽  
Transfer Functions ◽  
Representation Formula ◽  
Input Output ◽  
Numerical Approximations ◽  
Pseudo Differential Operators ◽  
Scattering Matrices

Acoustic waves travelling in axisymmetric pipes with visco-thermal losses at the wall obey a Webster–Lokshin model. Their simulation may be achieved by concatenating scattering matrices of elementary transfer functions associated with nearly constant parameters (e.g. curvature). These functions are computed analytically and involve diffusive pseudo-differential operators, for which we have representation formula and input-output realizations, yielding direct numerical approximations of finite order. The method is based on some involved complex analysis.

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