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S.B. Karavashkin

Differentiate (31) with respect to x

and with respect to y

Multiply (33) by i and subtract it from (32):

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

Proceeding the same with the rest derivatives, we yield

The equality (36) evidences that for the functions analytical after Caushy - Riemann, the total second derivative with respect to z retains its independence of the direction of its calculation and depends neither on ₁ nor on ₂.

Now express w( x, y) in (33) through u and v. It leads us to the following system:

and

It means that both their real and imaginary parts must satisfy Laplace equations.

Summing up, the carried out investigation shows that the theory of complex variables contains much wider scope. And a part of them can be extended to the vector algebra tool, to the theory of functions of several variables. And some results can be used even to analyse real functions of one variable. In this connection I hope that despite the simplicity of statements, this paper will be evaluated properly and it will gain a proper development, the same as other areas of mathematics.

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As an example, consider an application to solving the following second-order differential equation:

[3, p.61].

To simplify the solution, present z in polar form:

Substituting (23), (29) and (41) into (40), we yield: