SELF

81 - 83

S.B. Karavashkin and O.N. Karavashkina

This last has the cause that (5) is unable to indicate the equality of rates of tending w w₀, as the -vicinity is usually chosen after the most remote point w ( z ) corresponding to z and laying in the -vicinity of z₀ .

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And if we outline the real boundary of map z w , then, dependently on f ( z ) , it can take any complicated form (for example, () in Fig. 1).

The inconstant rate of tending w w₀  in relation to the direction of tending will cause the relation

to be dependent on the direction of tending w w₀ .

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It is known that (9) determines the total derivative of the complex function f ( z ) with respect to complex argument z that, as we can see it from the above analysis, is a complex analogue of the derivative over the direction in vector algebra.

To reveal the typical features of complex derivative, let us find the differentials of z and w.

To find the differentials of z, select in the complex plane z the -vicinity of the point  z₀ (see Fig. 2). Choose in this -vicinity a point z₁(x₁ , y₁ ), so that

Fig. 2. The diagram to calculate the increment z on the complex plane Z

We see from the graph in Fig. 2 that

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