86 - 88

S.B. Karavashkin

This derivation demonstrates also that the form of total differential does not depend on the way, how z is presented.

And finally, at = 0

If we abstract from the way, how the partial differentials dx and dy were compared, and substitute into (20) the limits taken from (11), we will obtain

i.e. the conventional value of the total differential. This fully corroborates the validity to identify dz with the total differential in the form (20).

Now determine the differential of w at the point w₀ at the condition that there

87

exists a one-valued mapping of the -vicinity of the complex plane z into the -vicinity of the complex plane w.

As we proved before, a function of a complex variable can be presented the most generally as

The increment in the function at the point w₀ takes the following form:

We see from (21) that w depends, as z, on one variable . Thus we can write

and

With the help of obtained differentials of an independent variable z and complex function w( z ), we can easy obtain an expression for the total derivative of a complex function. For it we will use (9), (20) and (22):

88

Now find the form of presentation of the total derivative of a complex variable with respect to z presented in the co-ordinate form, using (2). (23) takes the following form: