•Cauchy-Riemann-Fueter equations for regular and ψ-regular functions
and related boundary operators

We refer to [P1] and [P2] for the relevant definitions concerning regular, ψ-regular quaterninic functions, the Cauchy-Riemann-Fueter equations and the boundary differential conditions characterizing regular functions on a domain in C^2among harmonic functions.

CRF[{f _ 1, f _ 2}, z] computes the (left) Cauchy - Riemann - Fueter equations of f ... Overscript[z _ 2, _], ∂ f _ 1/∂ z _ 2 + ∂ Overscript[f _ 2, _]/∂ z _ 1} .

PsiCRF[{f _ 1, f _ 2}, z] computes the (left) Cauchy - Riemann - Fueter equations for (lef ... phic maps of two complex variables define a    ψ - regular function .

The following five differential operators will be used to give boundary differential conditions characterizing regular and ψ-regular functions on the unit ball in C^2among harmonic functions. Cf. [P1] for details.

DbarN[f, z] gives the normal part Overscript[∂ _ n, _] f = Overscript[z _ 1, _] &# ... ; Overscript[z _ 2, _] of Overscript[∂, _] f with respect to the unit sphere S .

L[f, z] applies the Cauchy - Riemann tangential (with respect to the unit sphere S) ... verscript[z _ 1, _] - z _ 1 ∂/∂ Overscript[z _ 2, _] to the complex function f .

L bar [ f, z ] applies the conjugate Cauchy - Riemann tangential (with respect to th ... ;/∂ z _ 1 - Overscript[z _ 1, _] ∂/∂ z _ 2 to the complex function f .

NFueter[f, z] applies the differential operator N = Overscript[z _ 1, _] ∂/∂ Overscript[z _ 1, _] + z _ 2 ∂/∂ z _ 2 to f .

TFueter[f, z] applies the tangential operator T = Overscript[z _ 2, _] ∂/∂ Overscript[z _ 1, _] - z _ 1 ∂/∂ z _ 2 to f .

RegularQ[{f _ 1, f _ 2}, z] tests for (left) Fueter - regularity of f = f _ 1 + f _ 2 j on the unit ball B . Here f is a function of z , Overscript[z, _] .

Psi RegularQ [ {f _ 1, f _ 2}, z ] tests for (left)    ψ - regularity ... the unit ball B .    Here f is a function of z , Overscript[z, _] .

Converted by Mathematica  (May 11, 2004)