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Some features of derivative of complex function

Fig. 1. Possible map of points z₁ and z₂ belonging to -vicinity of the complex plane Z into the points w₁ and w₂ belonging to the -vicinity of the complex plane W

Noting these definitions, consider some complex function f ( z )  that maps one-to-one the -vicinity of the point z₀   of the complex plane Z onto the -vicinity of the point w₀ of the complex plane W (see Fig. 1).

Choose in the -vicinity of the point z₀  two points z₁(x₁, y₁) and z₂(x₂, y₂) . In accordance with the definition of complex function, some points of mapping will correspond to them in the complex plane W: w₁(u₁, v₁)  and w₂(u₂, v₂). And according to the condition of one-valued mapping, if

(7)

Let us make the differences between the selected points z₁ , w₁,  z₂ , w₂ and z₀ , w₀  relatively:

81

Noting (7), in general case

At the same time

Thus, even from the condition

it does not follow generally that