Some relations among the classes of pseudoholomorphic functions. (English) Zbl 1151.30343

Çelebi, A. Okay; Koca, Kerim

Complex Variables, Theory Appl. 50, No. 7-11, 733-743 (2005).

In the paper under review, functions pseudoholomorphic in the sense of L. Bers are considered; a binary operation is introduced on the set \(E_{D_0}\) of generating pairs and some of its properties are given; this is followed by certain facts about the \(E\)-differentiation of pseudoholomorphic functions. The set \(E_{D_0}\) for a simply connected domain \(D_0\subset \mathbb C\) is defined to be \[ E_{D_0}:= E=(F,G)\colon\{ E\in H^1(D_0)\times H^1(D_0); \text{Im}(\overline{F(z)}G(z))>0, \forall z\in D_0\}, \] where \(H^n\) is the set of all differentiable Hölder continuous functions of order \(n\) with respect to \(x\) and \(y\) in \(D_0\) with \(z= x+iy\). The operation of \(*\)-multiplication is introduced as follows: given \(E_1=(F_1,G_1)\) and \(E_2=(F_2,G_2)\), then \[ E_1*E_2:=(F_1,G_1)* (F_2,G_2)= \frac12(F_1F_2-G_1G_2,F_1G_2+F_2G_1).\tag{\(*\)} \]
Denote by \(j\) the second imaginary unit of bicomplex numbers [see, for instance, D. Rochon and the reviewer, An. Univ. Oradea, Fasc. Mat. 11, 71–110 (2004; Zbl 1114.11033), and many other sources as well]: \(j^2=-1\), \(ij=ji\). Given two bicomplex numbers \(a+bj\) and \(c+dj\) (\(a,b,c,d\) are in \(\mathbb C\)), then \((a+bj)\cdot(c+dj)=(ac-bd)+(ad+bc)j\), which up to the coefficient \(\frac 12\) coincides with \((\ast)\).

Reviewer: Michael Shapiro (Mexico) (MR2155441)

MSC:

30G20

Generalizations of Bers and Vekua type (pseudoanalytic, \(p\)-analytic, etc.)

Citations:

Zbl 1114.11033