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S.B. Karavashkin and O.N. Karavashkina

4. Dynamical functions of complex variable

As one of important applications of analytical non-conformal mapping, we can refer to the mapping with the help of dynamical functions of complex variable. To illustrate, in Fig. 5 we show the dynamical analogue of the mapping shown before in Fig. 1:

where t is a time parameter, and    is the circular frequency of dynamical transformations of a complex function.

In Fig. 5 the Readers of SELF Transaction can see animated, and the Readers of Mathphys Archive can imagine that the time multiplier appearance in the right-hand part of (18) has led to the translational shift in time of the field lines of power in the plane W  into the field central region. With it the location of their prototypes in the plane Z  remains time-invariable, and therefore they still correspond to the stationary process. This feature just distinguishes the fields description by dynamical functions from the conventional description with which the function describing the power field characteristics depends directly on the studied domain co-ordinates. As the result, to describe completely the process in such fields, one needs using not one but two non-conformal mappings, both stationary and dynamical. The first of them introduces a one-valued analytical correspondence between the domain of the plane Z  and domain of values of the plane W . The second shows the power field time-transformation degree, so this is intended to describe the dynamical power function.