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On complex functions analyticity

Despite the necessity to use two mappings at the same time, the property of dynamical complex functions to describe the field power function in a plane enable us to solve the dynamical problems in complex dynamical fields which one meets often in hydro- and aerodynamics, electrophysics, acoustics, geophysics and so on. With it, naturally, the necessity of using the double mapping brings its peculiarities; we will try showing them by a simple specific example, determining the motion trajectory for a body having the mass m within the field of forces described by (18) and Fig. 5.

Suppose that at the initial time moment  t = 0   the body was located at the point w₀(x₀, y₀). Associate the stationary mapping between the planes Z and  W  with this moment. In our case this mapping will be described by (4). Since the problem is plane, suppose that the force field amplitude decrease occurs along the lines of force (which is very complicated to be taken into account by conventional methods in case of complex configuration of the lines of force) and is proportional to the first power of distance to the field centre, also along the lines of force.

Since in the considered problem the lines x = const  of the plane Z   correspond to the lines of force of the plane W, the attenuation degree is proportional to . The constant multiplier in the denominator was introduced to correlate the metrics of planes W and Z  along the field line of force. It is determined conventionally:

The force direction at each point is known to be determined by the direction of a tangent to the power line of force. Noting also that in our problem the field acts on the body towards the field centre, the tangent of force inclination  can be determined as follows:

This last transformation is caused by, in the considered problem in the plane Z   the equation  y = const corresponds to the lines of force. Substituting (18) into (20), we yield

where .

Taking into account the additional definition which we have made for the power field, we may write the differential equation of the body motion so

:where  K_F  is the coefficient determining the amplitude of field action on the studied body.

In (22) the dependence w(z)  corresponds to the stationary non-conformal mapping, because we completely took into account the features of the dynamics of process, when defined the power field additionally. At the same time, for a correct approach to solving (22) we have to note that despite the stationary pattern of mapping of  w(z) , the variation of a prototype co-ordinates in the plane Z will correspond to the time shift of the body in the plane  W. So in finding the solution of the differential equation (22) we have to note that the time dependence exists not only for the points of the body trajectory in the plane  W, but for their prototypes in the plane Z such dependence also takes place.