| V.1 | 84 - 86 |
Some pecularities of derivative of complex function |
|
|
|
| 84 | |
|
|
As
we see from (13), dz depends only on one real variable To show it, consider the triangle OAB (see Fig. 3) formed by the radius-vectors |
|
|
(14) |
In accord with the sine theorem |
|
|
|
whence |
|
|
(15) |
|
(16) |
In accord with the cosine theorem, |
|
|
(17) |
Substituting (17) into (16), we yield |
|
|
(18) |
85 Noting that |
|
|
|
yield |
|
|
(19) |
And from (19) we yield 86 |
|
|
(20) |
We see from this derivation that the form itself of writing the
total differential of dz with the fixed value of the direction of differentiation
(the angle |
|
Contents: / 77 - 78 / 78 - 79 / 80 - 81 / 81 - 83 / 84 - 86 / 86 - 88 /
, and at the same time it notes all partial differentials in 
-vicinity of z0 .
This property of the differential of z reveals the most visually when representing z
in polar form. 
) turns one of the independent
variables ( 
) into a non-differentiable parameter
depending only on the location of a point in which this total differential is sought, and
the second differential of d
z0 subtend direction, so this is
the differential in a common sense.