V.1

84 - 86

Some features of derivative of complex function

Fig. 3. The geometric construction, additional to Fig. 2, to calculate the increment z on the complex plane Z

As we can see from (13), dz depends only on one real variable and at the same time it accounts all partial differentials in the -vicinity of z₀ . This property of the differential z is seen best when we express z in the polar form.

Let us prove it. Consider the triangle OAB (see Fig. 3) formed by the radius-vectors

After the sine theorem

hence

After the cosine theorem

Substituting (17) into (16), yield

85

Noting that

yield

From (19) yield

86

It is well seen from the shown derivation that the very form of record of the total differential dz with the indicated direction in which we take the differential - the angle - turns one of independent variables - - into the non-differentiable parameter dependent only on the location of point at which we seek this total differential, and the second differential, d, does not depend on the direction of tending z z₀ , so it is not a differential in the trivial meaning.