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Some pecularities of derivative of complex function

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Noting the above definitions, consider some complex function f ( z ) proceeding the one-valued mapping of -vicinity of the point z₀ of the complex plane Z into -vicinity of the point w₀ of the complex plane W (see Fig. 1). Choose in the -vicinity of z₀ two points z₁(x₁, y₁) and z₂(x₂, y₂). In accord with the complex function definition, some points w₁(u₁, v₁) and w₂(u₂, v₂) of mapping on the complex plane W correspond them. And according to the condition of one-valued mapping, if

(7)

Form the differences between the picked points z₁ , w₁, z₂ , w₂ , z₀ and w₀ correspondingly:

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Noting (7), in general case

At the same time

Thus, even from the condition

generally there does not follow