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S.B. Karavashkin

This last is caused by the fact that (5) cannot note the equality of w w₀  subtend velocities, because the -vicinity is usually chosen with respect to the most distanced point w ( z ) corresponding to z and falling in the -vicinity of z₀ . But if

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we outline the real border of the mapping z w, then dependently on f ( z ) it can take any complex form (e.g., () in Fig. 1).

But the inconstancy of w w₀ subtend velocity dependently on the subtend direction causes that the relation

becomes dependent on the w w₀ subtend direction.

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As is known, (9) determines the total derivative of the complex function f ( z ) with respect to complex argument z. As we can see from this analysis, this function is a complex analogue of the derivative with respect to direction in vector algebra.

In order to reveal the salient features of the total complex derivative, determine the differentials of z and w.

To determine the differential of z, pick on the complex plane Z the -vicinity of the point z₀ (see Fig. 2). Pick in this -vicinity a point z₁(x₁ , y₁ ), and let

We see from the construction in Fig. 2 that

Tending z₁ z₀ and noting (10), we yield in the limit