110

S.B. Karavashkin and O.N. Karavashkina

Speaking of a body chaotical motion in the sink field, note, how the field lines of force effect on the direction of this motion. In the reverse changing of this motion (i.e., in the sink field transformation into the source field), the motion becomes well less chaotical, because in this case the force radial projection will be directed to the weak field. The construction of the Fig. 11 in which we show the body motion trajectory in the source field corroborate this. We see in this construction that in comparison with Fig. 7 the motion trajectories become much more smooth and the lines of force do not repulse the body.

At the same time, we see also that the trajectory still turns with the growing frequency, but now reverse. The same as it retains the transformation direction variation at = 1,0 s^-1  occurring at the same field frequency. This corroborates that the trajectory displacement pattern with the field increase is a general regularity of the body motion in such fields.

As we see, stationary and dynamical non-conformal mappings can be helpful in studying the processes in complex dynamical power fields of the most different nature. In this connection, the proved above definitions of analyticity of functions implementing the non-conformal mapping can serve a reliable tool helping to approach more accurately, and the main, more correctly to the studying the processes in which these functions are used.

In the carried out investigation we have revealed that the requirement, the limits to be equal for all sequences converging to the -vicinity of the point z₀, is excessive for the definition of differentiability of a function of complex variable. It suffices, the limits along all arcs regular in the -vicinity of the point z₀   and passing through  z₀  to exist.

General definition of differentiability can be presented in the form of two definitions - in general sense and after Caushy - Riemann correspondingly. The definition after Caushy - Riemann adds to the requirement, the limits for all arcs regular in the -vicinity of the point  z₀  to exist, the requirement, they to be equal. The condition of differentiability in general sense covers that after Caushy - Riemann. The class of functions differentiable in general sense covers that after Caushy - Riemann.

In accord with the broadening definition of the functions differentiability, the definition of their analyticity broadens too. The general definition of analyticity is presentable in the form of two definitions - in general sense and after Caushy - Riemann correspondingly. The condition of analyticity in general sense covers that after Caushy - Riemann. The class of functions analytical in general sense covers that after Caushy - Riemann.

To investigate the processes in complex dynamical fields, one needs to use two types of mappings, stationary and dynamical. This first serves to introduce the one-valued correspondence between the domain, where the function is determined, and the domain of values of non-conformal mapping, and the second serves to describe the pattern of the field power function variation in time. With it the structure of functions used for stationary and dynamical mapping must be the same.

In this studying we have revealed that when the body moves in complex central fields whose lines of force have a tangential component, with the growing in time frequency of transformation, the body trajectory displaces to the field centre (in case of a sink field) and vice versa (in case of a source field). Up to a definite frequency of the field, the trajectory displacement retains its direction even after it crosses the radial axis of the field. At the frequencies higher than that the displacement direction reverses.

As in case of source as in case of sink, if the field had a tangential component, under the field force action the body repulses to the periphery, to the weak field.

1. Karavashkin, S.B. Some peculiarities of derivative of complex function with respect to complex variable. SELF Transactions, 1 (1994), pp. 77-94. Eney, Ukraine, 118 pp.

2. Lavrentiev, M.A. and Shabat, B.V. The methods of theory of functions of complex variable. Nauka, Moscow, 1973, 736 pp. (Russian)

3. Vekua, I.I. Generalised analytical functions. Physmathgiz, Moscow, 1959 (Russian)

4. Volkovisky, L.I. Quasi-conformal mappings. L'viv University Publishing, 1954 (Russian)

5. Ilyin, V.A. and Poznyak, E.G. Foundations of mathematical analysis, part 1. Nauka, Moscow, 1971, 559 pp. (Russian)

6. Shilov, G.Ye. Mathematical analysis. Functions of one variable. Vol. 1-2 (two in one), Nauka, Moscow, 1969, 528 pp. (Russian)