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On complex functions analyticity

In the view of transformation of a path about the point z₀  with its mapping onto W, the contraction velocity gains the first-order importance, because just relation of the velocity of contraction to the point w₀ on the plane W to that to the point z₀ on the plane Z determines the value of a derivative along the picked out domain. With it, the different velocities in different directions do not mean yet the break, if it was (and for analytical function must be) a smooth function with respect to angle  _z. This is just our task to find the conditions which might select from multitude functions the classes having the above properties.

For it, consider in a plane Z some small -vicinity of the point z₀. In this vicinity give parametrically some arc x(), y()  passing through the point z₀. Then for the function which maps the -vicinity of z₀  into - vicinity of the point w(z₀), we can write in the most general form so:

In other words, we presented the studied function w(x, y) as a complex function with respect to . Now if we differentiate this function with respect to , we yield

In its turn,

We can see from (15) and (16) that, if the chosen arc (or rather a set of arcs) was regular in the studied - vicinity, then in (15) all particular derivatives x and y with respect to exist and general differentiability of the function w(x, y) is defined by the existence of particular derivatives with respect to x and y. In this case it suffices simply to use the definition of the existence and continuity of the particular derivative, to substantiate the differentiability of the function w(x, y) itself. But if the arc was irregular, one cannot conclude, is or is not this function differentiable. This proves that the regularity of an arc in the -vicinity of the point   z₀ offers one to find, is the function w(x, y) differentiable along the given arc (or a family of arcs).

Noting the proved, to make the function differentiable in -vicinity of z₀, it is sufficient, it to be differentiable for any regular arcs in the -vicinity of the point  z₀. If in the -vicinity there is at least one arc along which the function w(z₀) is non-differentiable, then naturally, this function on the whole cannot be thought differentiable along the selected direction (though it can be thought a partially differentiable along the selected direction or in the sector). And vice versa, if a regular arc along which the function w(z₀) is non-differentiable is absent, then any families of curves being smooth in the -vicinity of z₀   map one-valuedly into the families of w(z₀) being smooth in the -vicinity. Noting the proved statement of regular arcs, it speaks of the full differentiability of w(z₀) at the point z₀.

On the basis of the carried out investigation and taking into account the conventional definition 2, we can formulate the definition of general differentiability so:

Definition 4. The function of complex variable w = f(z) is differentiable in general sense at the point  z = a , if the limit

existed at  z = a for any regular arc in the -vicinity of the point  z = a  passing through the point  z = a.

We can easy make sure that the conventional condition of the function differentiability after Caushy - Riemann is a particular case of definition 4 with additional condition, the limits along all regular arcs to be equal.

Having defined the conditions of differentiability in general sense, we in fact generalised the conditions of functions analyticity in general sense, because, according to the conventional definition 1, the differentiability condition is the principal criterion of the function analyticity. So, following the division of the differentiability definition into that general and that after Caushy - Riemann, the function analyticity can be also defined now in general sense and after Caushy - Riemann. With it the class of functions analytical in general sense will be naturally covering for that analytical after Caushy - Riemann. And the analyticity definition itself will remain invariable, with accuracy to the refinements connected with the function differentiability.

Finally, our new definition of complex-variable functions analyticity can be easy extended from the -vicinity of the point z₀  to some connective domain of the plane Z. And it does not require a new proof, because we did not transform the concept of both  - and -vicinities. In this connection, "the function is analytical in an open domain D, if it was analytical at each point of this domain" [1, p.197].

Note however that far from every extension of the analyticity concept in general sense will be so simple. In particular, there will be problems with the analytical continuation through the border. But this is the subject of another large investigation.