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S.B. Karavashkin and O.N. Karavashkina

Differentiating (31) with respect to x, we yield

Differentiating (31) with respect to y, yield

Multiplying (33) into i and subtracting it from (32), yield

Substituting in (30) the terms ²w/y² and ²w/xy   in accordance with (32) and (34), yield

Similar operations concerning other derivatives give

The equality (36) evidences: for functions analytical after Caushy – Riemann, the total second derivative over z also retains its properties to be independent of direction in which it is taken and of  ₁ and ₂ . If we express w( x, y) in (36) through u and v, we will come to the system of inequalities

and

that are the Laplace equation for two variables. One more property of centrally symmetrical functions of complex variables follows from it:

i.e., both real and imaginary parts have to satisfy the Laplace equation.

Summing up this study, we would say, its results visually show much wider scope of theory of functions of complex variable and a part of this scope can be transferred to the tool of vector algebra, to the theory of functions of several variables. And some results can be used even for the analysis of real functions of one variable. In this connection, I would hope that despite all simplicity of the shown computations, the Readers will understand the importance of this work and it will gain its development, just as other areas of mathematics.

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To finish with, let us consider as an example the application of above material to the solving of the second-order differential equation

[3, p. 61]. To simplify the solving, represent z in the polar form

Substitute (23), (29) and (41) into (40) and yield